A dissertation under the direction of Gordon Simons.
SOME STRUCTURAL RELATIONSHIPS BETWEEN WEAK CONVERGENCE
OF PROBABILITY MEASURES AND CONVERGENCE IN PROBABILITY
Flav;o W. Rodr;gues
Department of Statistics
University of North Carolina at Chapel HiH
Institute of Statistics Mimeo Series No. 812
March, 1972
TABLE OF CONTENTS
PAGE
CHAPTER
ACKNm~LEDGMENTS
iii
ABSTRACT
I
v
INTRODUCTION
1.1 Introduction and preliminary background
1.2 Review of the literature and description of the
resul ts
II
9
18
24
GENERALIZATION TO COMPLETE, SEPARABLE METRIC SPACES
3.1 The Skorokhod partition of S
3.2 The non-atomic case
3.3 The atonric and the general case
IV
4
THE REAL LINE
2.1 The non-atomic case
2.2 The atomic case
2.3 The general case
III
1
SOME RELATIONSHIPS BETWEEN THE METRICS LAND
4.1 The lower bound for p
4.2 Weak convergence and equivalent probability
measures
32
49
61
p
67
77
APPENDIX
86
REFERENCES
91
ACKNOWLEDGMENTS
I would like to express my deepest gratitude to my advisor
Professor G. D. Simons for proposing this problem and for the many valuable suggestions he made during the course of this investigation.
I
would also like to thank him for his patience in going through the several first drafts of the manuscript and for his words of encouragement
and confidence in the difficult moments.
I wish to thank Professor W. Hoeffding, Professor M. R. Leadbetter
and Professor W. L. Smith for reading the manuscript and offering helpful suggestions.
I extend my thanks to those members of the faculty in the Departments of Statistics and Mathematics who contributed to my education at
the University of North Carolina at Chapel Hill.
For financial support, I would like to express my gratitude to
Conselho Nacional de Pesquisas, (C. N. Pq.),
Brazil, which granted me
a scholarship for the larger part of my stay in the United States.
My thanks also go to Pan American Health Organization for a travel
grant and financial support during my first year of graduate studies.
It is a pleasure to acknowledge the cooperation and understanding I
received from the faculty and administration of the School of Public
Health of the University of SAo Paulo during my leave of absence.
lowe a deep sense of gratitude to my wife, Regina, for her continuous support and enthusiasm during all the phases of my studies.
also want to thank Dr. Carlos A. B. Dantas whose insistence and
I
iv
enthusiasm were greatly responsible for my decision to come to the
United States to pursue graduate studies.
Finally, I thank Mrs. Cynthia Grossman for her able and careful
typing of the manuscript.
ABSTRACT
Let
{X}
n
bility space
distributions
Q.
(Q,
A,
{Qn}
P)
whose corresponding sequence of probability
is weakly convergent to a probability distribution
In this study, we show the existence, on
{Y}
n
(Q,
A,
P),
of a sequence
of identically distributed random variables, with probability dis-
tribution
n
be a sequence of random variables defined on a proba-
Q,
such that
tends to infinity.
Xn - Yn
converges to zero, in probability, as
In the special case of a non-atomic
(Q,
A,
P)
we use the quantile function, associated with the distribution function
of
Q,
to explicitly determine a particular version of the
functions of the
X 's
n
Y 's
n
as
and of an auxiliary uniform random variable.
This result is then extended to sequences of random elements taking
values on a complete, separable, metric space.
To accomplish this ex-
tension we consider a total ordering of the metric space induced by a
special sequence of partitions which were first introduced by Skorokhod.
The techniques used in the real line are adapted to metric spaces by
means of a type of generalized distribution function associated with
each probability measure on the space.
Some relationships between the Inetric
p,
associated with conver-
gence in probability, and the Levy-Prohorov metric,
vestigated.
L,
are also in-
In particular, a result relating convergence in probability,
weak convergence and the class of probability Ineasures which are equivalent to
P,
is shown to be valid on separable metric spaces.
CHAPTER I
INTRODUCTION
1.1.
Introduction and preli8inary background.
The basic facts, concerning the extension of the theory of weak
convergence of probability measures, to metric spaces, have been known,
at least, since 1940 (see, e.g. Alexandrov [1940]).
However, it was not
until 1956, with the publication of Prohorov's fundamental paper: "Convergence of random processes and limit theorems in probability theory",
that the importance and far-reaching consequences of such an extension,
were fully understood.
A detailed account of both the theory and appli-
cations of weak convergence, plus an extensive bibliography, can be
found in the monographs by Parthasarathy [1967] and Billingsley [1968].
In this dissertation, we will be concerned with problems relating
weak convergence with other types of convergence, and with the structure
of the basic probability space,
(n, A, P).
we are given a weakly convergent sequence,
Suppose, for example, that
{P },
n
of probability mea-
sures on the a-field of Borel sets, of a separable metric space
Consider, now, a probability space,
define a sequence,
{X},
n
(n, A, P),
where it is possible to
of random elements, with values in
corresponding sequence of probability distributions is
{p }.
n
interested in the implications of the weak convergence of the
the convergence properties of the
Xn 's
S.
S,
whose
We are
Pn 's
and for the structure of
for
2
(n, A, P).
Before describing our results, we need to recall some defi-
nitions and facts, about probability measures in metric spaces.
From now on,
distance
d.
S will always denote a separable metric space with
The particular distance will, in general, be of no concern
S
to us; what matters is that the topology of
metric.
is given by means of a
For the reader's convenience we have assembled, in the Appendix
the results about metric spaces that will be needed in the sequel.
Definition 1.1.1.
The Borel a-field of
containing the open (closed) sets of
since
S
balls of
to
(S, S)
(n, A,
P)
(S, S).
A sequence
{Q}
of probability measures on
n
Q on
for every real valued, continuous and bounded function,
f fd~
~
Definition 1.1.4.
A Borel set
Here,
f fdQ
Let
as
n
~
S)
(S,
f,
on
S,
if
we
~.
Q be any probability measure on
A is said to be a Q-continuity set if
oA
Any
induces, in the usual
is weakly convergent to a probability measure
(1.1.1)
S.
will be called a random element (r.e.) ·of
Definition 1.1.3.
zero.
Sand,
A measurable function from a measurable space
way, a probability measure on
S)
It will be denoted by
S.
r.e. defined on a probability space
(S,
is the smallest a-field
is separable, it is also the a-field generated by the open
Definition 1.1.2.
(n, A)
S.
S
denotes the boundary of
A.
Q(oA)
(S,
S).
is equal to
3
THEOREM 1.1.1.
(5, S).
Then~
verges to
Pn
Let
P be probabiUty measures on
{Pn } and
aonve~es
weakZy to
P if and onZy if
for every P-aontinuity set A
P(A)
aon-
Pn (A)
S.
E
For the proof and for other conditions equivalent to weak convergence, see Billingsley [1968; pages 11-12].
For any subset
...
(1.1. 2)
A of
5
and
{xE5: d(x,A)
Definition 1.1.5.
Let
P
L(P,Q)
=
and
and
inf{€ > 0: P(F)
L is a metric on the space
Z(5)
let
Q be two measures on
P
for all
0,
>
oJ •
$
Levy-Prohorov distance, between
(1.1.3)
0
The
Q is defined to be:
$
F,
S.
Q(F€)+£,
closed,
F c S}.
of probability measures on
(5, S)
and it has been shown, by Prohorov [1956] and Dudley [1968], that convergence in the metric
L
For random elements
is equivalent to weak convergence.
X and
Y,
from
(n, A, P)
(5, S)
we
will denote by
d(X,Y),
d(X(w), Y(w».
It can be shown, (see Billingsley, [1968], pg. 225) that
when
5
is separable,
d(X,Y)
Definition 1.1.6.
(5, S).
all
the function that at
to
Let
WEn
takes the value:
is a random variable.
{X}
n
and
X be r.e.'s from
The sequence is said to converge in probability to
£ > 0:
P({w: d(X ,X)
n
~
£})
converges to zero as
n
(n, A, P)
X,
if, for
tends to
infinity.
Convergence in probability can also be expressed by means of a
metric, on the space of all r.e.'s from
(n, A, P)
to
(5, S).
to
Given
4
any two r.e.'s
(1.1.4)
X and
p(X,Y)
=
Y define
inf{£ > 0: P({w: d(X,Y) > £})
S
£}.
If we interpret equality of r.e. 's to mean equality a.e. then,
is a metric and convergence in the metric
p
p
is equivalent to conver-
gence in probability.
1.2.
Review of the literature and description of the results.
We start by quoting the classical result which, loosely speaking,
says that convergence in probability implies weak convergence.
THEOREM 1.2.1.
with values on
Let
n
(8, S).
corresponding sequence
and X be r.e. 's defined on
{X}
{X } converges in probability to
n
{Qn}
the
X~
of probability distributions converges
Q of
weakly to the probability distribution
true in the case where
A, P)
(Q,
x.
The converse is also
Q has aZl its mass concentrated on a singleton.
In this dissertation we shall look into several types of converses
for Theorem 1.2.1.
As a matter of fact, our main results could be con-
sidered as converses for Theorem 1.2.1, in the special case where the
r.e. 's are assumed to be defined on the same probability space.
The
following theorem, due to Skorokhod [1956], can be regarded as an
existence type of converse for Theorem 1.2.1 and will playa fundamental
role in the sequel.
THEOREM 1.2.2. (Skorokhod)
sures on
(8, S)
and assume that
Let
{p}
n
{p}
n
and P be probability mea-
converges weakly to
is separable and compZete 3 we can find a probability
8pace~
P.
If S
with random
5
eZements
a)
and X defined on
{X}
n
For each n;:;; I,
it~
such that
is the pl'obabiZity distribution of xn
Pn
and P is the pl'obabiZity distribution of X.
b)
converges to
Xn
X
everywhel'e~
as n
tends to infinity.
This result has been extended, by Dudley [1968], to spaces
which are only separable.
8
For further extensions, which don't assume
separability see the paper by Wichura [1970].
A survey paper by Pyke
[1970] gives several examples of a.s. convergent processes, constructed
with the help of Theorem 1.2.2, which have a wide variety of app1ications to probability and statistics.
For
if
P
tance
S complete, Strassen [1965] proved the important result that
and
Q are any probability measures in
L(P,Q)
Z(8),
is the minimum distance, "in probability", (see Defini-
tion 1.1.4) between r.e.'s distributed according to
this result, it follows that if
of r.e.'s
and
P,
{X}
n
and
{y},
n
P
n
n
P
and
converges weakly to
P,
Q.
From
sequences
whose distributions are respectively,
could be constructed in such a way that,
to zero as
the Prohorov dis-
tends to infinity.
{P}
n
converges
n
n
Of course, this last result is also
p(X , Y )
implied by Skorokhod's theorem which is in fact, stronger.
Both results mentioned above, assume as given a weakly convergent
sequence,
{P },
n
of probability measures and go on to construct a prob-
ability space, with a sequence of r.e.'s defined on it, distributed according to
{P }
n
and satisfying a specific convergence property.
In
our main result (Theorem 1.2.3, below) we take a different approach,
assuming both the probability space and the sequence of r.e. 's to be
given a priori.
6
THEOREM 1.2.3.
S.
distributions~
Q.
Then~
there
be a sequence of r.e. 's defined on a
{X }
n
(n, A, P),
probability space
metric space
Let
with values on a
complete~
separable
Suppose that the corre sponding sequence of probabi Zi ty
{~}~
exists~
's~
distributed r.e.
converges weakly to the probability distribution
on
(n, A,
P)~
a sequence
{y}
n
Q~
with probability distribution
distance "in probabi Zi ty"
p (x , y )
n
n
of identically
such that the
converges to zero as n tends to
infinity.
Remarks: 1)
admissible for
We say that a probability measure
(n, A, P)
a r.e., with values in
S,
Q,
on
(8, S),
is
if there exists, in this probability space,
whose probability distribution is
Q.
It
follows, from Theorem 1.2.3, that the class of probability measures on
(S, S),
which are admissible for a given probability space, is closed
under weak convergence.
2)
Suppose that for some probability space
some probability measure,
(n, A, P)
alence) r.e. on
Then, i f
{X }
n
Q on S,
whose probability distribution is
is any sequence of r.e. 's on
(n, A, P),
Q,
Q.l
whose proba-
it follows that
X
n
X in probability.
In Chapter II, we discuss the case where
the usual metric.
1
and for
X is the unique (up to an equiv-
bility distributions are weakly convergent to
converges to
(n, A, P)
S is the real line with
In this case, the proofs can be greatly simplified by
An example of this situation is provided by a degenerate probability measure, i.e. a probability measure that put all its mass on a
singleton. For spaces with atoms (see Definition 2.1.2) this situation
may occur even for non-degenerate Q.
7
the well-known constructions based on the natural order of the real
line.
For example, we will show that if
with distribution function
F,
X is any random variable,
defined on a non-atomic probability
space, (Definition 2.l.3) it is possible to modify
obtain a uniform random variable
U,
pressed as a measurable function of
F(X}
in such a way that
U.
in order to
X can be ex-
This result, described in
Theorem 2.1.2, may have some interest in itself.
For atomic probability
spaces, we show that Theorem 1.2.3 is a consequence of the relations between tightness (see Definition 2.2.2) of the sequence of probability
measures and the existence of a.e. convergent subsequences. (Theorem
2.2.2}.
The general case requires a little more than a simple combi-
nation of the other two, in virtue of the fact that weak convergence is
not, necessarily, preserved under conditioning.
In Chapter III, we consider the problem in an abstract complete,
separable, metric space.
The difficulty here will be in the non-atomic
case, since the powerful machinery of distribution functions is not
available.
We will show, however, that an equivalent procedure can be
developed, by partitioning the metric space, in the way used by
Skorokhod in the proof of his result (Theorem 1.2.2).
In Chapter IV, we will discuss some relationships between the metrics
Land
bility space
p
from the point of view of the structure of the proba-
(n, A, P).
By restricting
L to the subspace of admis-
sible probability measures we will be able to discuss a structural version of Strassen's result.
Furthermore, we will extend to metric spaces
a result of Padmanabhan [1970] and show how this extension will allow us
to use
L to define a new metric
equivalent to
p.
PO'
in the space of r.e.'s, which is
By means of an example we will show that
Po
is not
8
necessarily complete which implies that, although equivalent, the two
metrics generate different uniformities.
CHAPTER II
THE REAL LINE
In this chapter we consider sequences of probability measures on
B of Borel sets of the real line
the a-field
1
(R , B)
known, a probability measure on
1
(R ).
As it is well-
is completely determined by
the corresponding distribution function (to be hereafter abbreviated as
d.f.).
A random variable (r.v.),
(n, A, P)
induces in
PX(B) = P(X
-1
(B»
1
(R ,
for all
X,
defined in some probability space
B) a probability measure Px given by:
B
€
B.
We will make use of the words
inc~asing~ dec~asing~ positive~
negative in their loose interpretation.
added when necessary.
The qualifier strictly will be
All d.f. 's are assumed to be continuous from the
right and proper, that is such that
F(-oo) = 0
and
Finally, we recall that the weak convergence of
F(+oo)
{p}
n
= 1.
to
Pis,
here, equivalent to the convergence of the corresponding sequence of
d.f.'s,
{F },
n
to the d.f.
F
of
P,
at all points of continuity of
the latter.
2.1.
The non-atomic case.
Definition 2.1.1.
A
€
A,
we define the
Let
(n, A, P)
be a probability space.
P-equivalence class
determined by
A,
Given
to be:
10
[AJ
= {B
€
A:
P(AAB)
= a},
Definition 2.1.2.
where
6
B
Remapk.
If
€
A
for which
A
€
of
X,
A is an atom of
(n, A, P)
{X
X is a r.v. defined on
-1
(n, A, P)
({x}): x is a jump point of
P-equivalence cZass [AJ.
(n,
P(A6B)
F is the d.f.
-1
are
the event A and the
Statements such as: the event A is an atom,
= O.
A and
Bare
Since the intersection of two distinct
A, P) has probability zero it follows that a probability
(n, A, P)
and put:
Definition 2.1.3.
atomic.
On the
F}.
space has at most a countable number of atoms.
atoms of
and such
(n, X (B), P)
can be made rigorous by the convention that 2 atoms
atoms of
0
peA).
~
and
We will not, in general, distinguish between
equal whenever
>
then, the d.f. of every
then the atoms of the probability space
given by:
peA)
is
A, B c A, we have either PCB) = 0 or PCB) = peA).
r.v. defined on this space has a jump point of size
other hand if
(n, A, P)
An atom of a probability space
the P-equiva1ence class of a set
that for all
indicates the symmetric difference.
AO
~
= Un= 1
Let
{A}
n
denote the
An •
A probability space without atoms is called non-
A probability space is called atomic if
P(AO)
= 1.
The following theorem is due to Varadarajan [1958]:
THEOREM 2.1.1.
If
(n, A, P)
it is possible to define r.v.'s
is a non-atomic ppobability space,
~1'~2"'"
with aPbitrapy, pre-
assigned consistent, joint distPibutions.
CopolZ~.
A probability space admits a uniformly distributed ran-
dom variable if and only if it is non-atomic.
11
Definition 2.1.4.
Let
F
be any d.f.
inf{x
€
R: F(x) ~ t}.
For all
t,
0 < t < 1,
define:
(2.1.1)
1
F- (t)
=
It is easy to see that
-1
F
F-1 ,
Some properties of
is increasing and continuous from the left.
that will be needed later, are listed in the
following lemma:
Lemma 2.1.1.
o<
t < 1,
Let
F
be any d.f.
For all real
x
and every
t,
we have:
and
i)
ii)
x < F-1 (t)
if and only i f
implies
iii)
<
t
iv)
F
F(x)
-1
t
::;
F(x)
F- 1 (t) ::; x.
implies
is continuous at
t
if and only if
either empty or a singleton.
at
x
and
if and only if
{t: F
{x: F(x)
Conversely,
-1
(t)
F
= t}
is
is continuous
= x} is either empty or a
singleton.
v)
-1
converges weakly to
to
F
n
Proof:
F
F
-1
F
-1
if and only if
at all continuity points of
F
-1
Fn
converges
•
i, ii, iii, iv are direct consequences of the definition of
and the properties of the infimum of a set of real numbers.
prove
v, suppose first that
note a continuity point of
F
n
F
-1
•
converges weakly to
Let
In correspondence to a given
arbitrary, choose two continuity points,
(2.1.2)
F.
and
x
and
y,
of
F,
To
dee: > 0,
such that:
12
(2.1.3)
y - x
<
E.
This choice is possible in virtue of the fact that the set of con tinuity points of
F
is dense in the reals.
By part ii of the lemma, we have:
(2.1.4)
F(x) < to'
By part i of the lemma and the fact that
F
is increasing, we have:
(2.1.5)
Finally, since
to
-1
F
is a continuity point of
,
the first half of
part iv of the lemma implies:
(2.1.6)
to < F(y).
The convergence of
F (x)
n
to
F(x)
F (y)
and of
to
n
F(y)
with (2.1.4) and (2.1.6) imply that there exists an integer
together
N,
such
that:
(2.1. 7)
for all
n
~
N.
Parts ii and iii of the lemma, imply:
for all
(2.1.8)
n
~
N.
By putting together (2.1.2), (2.1.3) and (2.1.8) we have:
for all
(2.1.9)
This completes the proof of the first part of
n
~
N.
v.
To prove the other half, observe first that the set of continuity
points of
F
-1
is dense in
(0,1) .
Hence, given
E
> 0,
and a
13
continuity point
o
a)
b)
o of F, consider the possibilities:
X
< F(x ) < 1.
Choose two continuity points,
O
-1
of
F
such that:
If
F(x )
O
,
=0
t
1
F(x )
O
or
tinuity point
t
O
=1
t
and
2
and
1
- t1 <
2
t2'
E.
choose, respectively, a con-
or a continuity point
< E
1
< F(x ) < t
t
t
2
> 1 - E.
o
The rest of the proof is analogous.
Remark.
It follows from part iii) of the lemma that if
uniform r.v. on
[0,1]
and
F
U is a
1
is any d.f., the r.v.
F- (U)
has d.f.
F.
Hence, it can be shown that if
X is a r.v. with a continuous d.f.
F,
there exists a uniform r.v.
such that:
fact, since
ting
U
F
= F(X)
fact that
is continuous,
X and
1
F- (F(X»
Let
(n, A,
U~
a uniformZy distributed r.v.
Remark on notation.
(2.1.10)
F(t).
=
Proof of the theorem.
2.1.1.)
real numbers,
(2.1.11)
a.e.
In
defined in a non-
=
(n, A,
Then, there exists in
such that:
F
1
F- (U)
and
x
E
P)
= X a.e.
1
R ,
we write:
G,
we write:
Choose a uniformly distributed r.v.
Z
Z
in
is assured by the Corollary to Theorem
G the joint d.f. of
(x, z),
H(x,z)
P).
F~
with d.f.
lim G(t, y).
t x
(The existence of
Denote by
r.v.~
Similarly for any bivariate d.f.
G(x-, y)
A, P).
=X
(U)
is uniformly distributed and put-
For any d.£.
t
(n,
-1
have the same distribution.
X be a
atomic probabiZity space
= 1imttx
F(X)
F
the result follows from part i) of the lemma and the
THEOREM 2.1.2.
F(x-)
U
X and
define:
F(x-) + G(x,z) - G(x-,z).
Z.
For every pair of
14
Now, define for
(2.1.12)
=
U(w)
WEn:
H(X(w), Z(w».
It follows that:
(2.1.13)
F(X(w)-)
s
U(w)
We claim that the r.v.
s
F(X(w»
all
w
€
n.
U satisfies the requirements of the theorem.
Recall that to show that
U is uniformly distributed, it will be enough
to show that:
(2.1.14)
Let
P({w: U(w) < t})
t,
0
<
t < 1,
s
t
be given.
s
P({w: U(w) s t})
Put:
all
t E (0,1).
-1
o = F (t). By part i) of
X
Lemma 2.1.1, we have:
(2.1.15)
=
t.
We first prove that:
(2.1.16)
{w: U(w)
<
c
{w:
W
Now, observe that for any real
~
t.
X(w) s xo}.
o is such that: U(W O)
F(X(w )-) < t.
o
In fact, suppose that
(2.1.13), that:
t}
y > xO'
<
t.
we have:
It follows, from
F(y-)
~
F(x o )
~
The result follows.
We now consider, separately, the two possible cases in (2.1.15).
Case 1.
F(x )
O
= t.
First note that, X(w) s xo'
together with (2.1.13)
imply:
U(w)
Hence, we have
S
F(X(w»
S
F(x O)
=
t.
15
(2.1.17)
~
{w: X(w)
xo}
~
{w: U(w)
c
t}.
(2.1.16) and (2.1.17) imply:
~
P({w: U(w) < t})
Since,
F(x
Case 2.
F-1
o) = t,
= P(x O)
F(x -)
O
t.
$
t.
>
$
t}).
This assumption and the definition of
Hence:
(2.1.18)
<
Now, observe that,
uous function of
mum is zero.
P({w: U(w)
$
(2.1.14) follows.
F(p-l(t))
imply:
F(x )
O
G(xO'z) - G(xO-'z)
is,
whose maximum is
Z,
for
z
[0,1],
€
F(x ) - F(x O-)
O
a contin-
and whose mini-
Since a continuous function, defined in a compact set,
assumes all the values between its maximum and its minimum, it follows
from (2.1.18), that there exists
zo
€
[0,1],
such that:
(2.1.19)
We will show that:
(2.1.20)
{w: U(w)
{w:
<
t}
Z(W)~zo}}
c
{w: X(w)
<
xo}
c
{w:
U(w)
$
t}.
{{w: X(w)=xO}n
u
To prove the first half of (2.1.20) assume that
U(w
o)
< t.
we are done.
It follows by (2.1.16), that
Suppose:
X(W ) = xo.
O
X(W O)
Wo
$
is such that:
xo.
U(w )
O
~
t
X(W O) < xO'
Hence, from the definition of
we have:
From the assumption:
If
and (2.1.19), we have
U,
16
Observe now that:
increasing with
G(xo'z) - G(xO-'z) = P({w: X(w)=xo}n{w:
z,
Z(w)~z})
is
and hence (2.1.21) implies:
(2.1.22)
The first half of (2.1.20) follows.
To complete the proof of (2.1.20), let
X(w 1 ) ~ xo.
If
X(w1 ) < xo'
w
1
be such that:
Xo = F- 1 (t),
since
we have, by part i)
of Lemma 2.1.1, that:
(2.1.23)
(2.1.13) and (2.1.23) imply:
now, that:
X(w 1 ) = X
o
and
Z(w )
1
U(w )
1
~
z00
<
t
and we are done.
Suppose
We have:
= F(X O-) + G(x O'Z(w1 » - G(xO-'Z(w1 »
~
F(x O-) + G(xo'zo) - G(xO-'zo)'
From (2.1.19), the expression on the right hand side is equal to
t
and
the proof of (2.1.20) is completed.
Using (2.1.20) we can write:
(2.1.14) now follows from (2.1.19) and the proof that
U has the uni-
form distribution is complete.
To finish the proof of the theorem, observe that
whose d.f. is
F.
-1
(U)
is a r.v.
On the other hand, from (2.1.13) and part i) of
Lemma 2.1.1, we have:
(2.1.24)
F
17
Since
1
F- (U)
and
X have the same distribution, (2.1.24) implies:
o
Remarks.
1) In the case that
F
is continuous, the theoremre-
duces to the situation discussed in the remark following Lemma 2.1.1.
2) The r.v.
is clearly not unique and the role played by
U
U,
in the construction of
Z,
would have been fulfilled by any r.v., with
a continuous d.f., taking its values on a compact set.
THEOREM 2.1.3.
Let
n
non-atomic probability space
sequence,
be a sequence of r.v. 's defined in a
{X}
A, P). Suppose that the corresponding
(n,
of d.f. 's is weakly convergent to a d.f.
{F},
n
(n, A, P)
there exists in
F,
buted r.v. 's, with d.f.
a sequence,
F-1 (U)
r. v. ,
n
converges to
-1
F
n
to
F
-1
= O.
D})
€
F
-1
to
F
n
F- 1 (U)
Therefore,
n
n
n
=
u ,
n
X a.e.
n
.
such that:
all
n ~ 1.
Define:
(2.1.26)
implies the convergence of
F,
-1
F
Y
n
=
.
Since the set
D of
is, at most, countable it follows that:
uniformly distributed r.v.
F- 1 (U)
con-
In fact, by part v) of Lemma
By the previous theorem, to each r.v.
(2.1.25)
Xn - Yn
=
is any uniformly distributed
at all points of continuity of
discontinuities of
P({w: U(w)
U
F-1 (U) a.e.
2.1.1, the weak convergence of
Zn
tends to infinity.
n
We first prove that i f
Proof·
Then,
of identically distri-
{Y},
n
such that the sequence
verges in probability to zero as
F.
F- 1 (U)
n
all
n ~ 1.
converges to
X,
n
F
-1
(U) a.e.
we can associate a
18
Clearly t the
y 's
are identically distributed with d.f.
n
more t observe that for fixed
nt
U
n
F- 1 (U) - F- 1 (U).
the distribution of
Further-
1
1
F- (U) - F- (U )
n
n
n
the distribution of
depends only on the distribution of
F.
and is, therefore, the same as
Since the latter sequence con-
n
verges to zero a.e. it follows that:
verges to zero, in probability, as
Remark.
conn
In the case where all the
we have, for all
n
~
1,
U
n
= Fn (Xn ).
X 's have continuous d.f.'s,
n
=
F-1 (F (X»
n n
y
Hence, each
itly determined as a Borel measurable function of
Y
n
o
tends to infinity.
all
n
can be explicalone Le.:
X
n
n ~ 1.
2.2. The atomic case.
We will now consider sequences of r.v.'s defined in an atomic probability space.
Many of the basic definitions and results, used in this
chapter, will again be needed in Chapters Three and Four in the more
general context of separable metric spaces.
their particular versions for
R1 ,
We introduce them here, in
in order to make this chapter com-
plete and self contained.
Definition 2.2.1.
Let
G be a family of d.f.'s (equivalently a
family of probability measures on
(R
1
,8».
We call
G relatively com-
paot if every sequence of elements of G contains a subsequence which
is weakly convergent to a d.f. (not necessarily in
Definition 2.2.2.
for every
that:
E >
0,
A family
G).
G of d.f.'s is said to be tight if
there exists a closed, finite interval,
[a,b],
such
19
(2.2.1)
F(b) - F(a)
1 -
>
all
£
THEOREM 2.2.1. (Prohorov)
F
G.
€
G of d.f. 's is relatively
A family
compact if and onLy if it is tight.
For the proof and a discussion of the implications of this result
see e.g. Billingsley [1968, pages 35-40].
THEOREM 2.2.2.
Let
be a sequenae of random variables de-
{X}
n
fined in an atomia probability spaae.
{F }
n
of d.f. 's is
tight~
If the aorresponding
then every subsequence of
sequenae~
has a further
{X }
n
subsequence which converges a.e.
Proof.
Since every subsequence of a tight sequence of d.f.'s is
itself a tight sequence it is enough to show that
convergent subsequence.
(n, A, P)
{X}
n
{Ai}i~l
Let us denote by
has an a.e.
the atoms of
and write:
P(A.)
=
1.
all
i
~
1.
Definition 2.1.2 and the assumption that the space is atomic, imply:
00
(2.2.2)
Pi > 0
all
i
~
1
L
and
i=l
Furthermore, for every fixed
{x}
ni n~l
(2.2.3)
i
1
~
Pi
= 1.
there exists a numerical sequence
such that:
Xn(W) = x ni
for all
n
~
1
and almost all
We now prove that, for every fixed
sequence of real numbers.
In fact, since
i
~
1,
W € Ai'
{xni}n~l
Pi > 0,
is a bounded
we can choose
£,
20
such that:
(2.2.4)
o<
8
<
p.,
1
It follows from the assumption of tightness that there exists a
bounded, closed interval,
[a,b],
such that:
(2.2.5)
all
n
~
1.
On the other hand, from (2.2.3) and (2.2.4) we have:
(2.2.6)
P({w: Xn (w)
= xi})
n
It is clear that if for some
have a contradiction.
=
nO
~
1,
we had
It follows that
of real numbers, for all
i
~
all
x
no
{x}
ni n~l
i
t
[a,b],
n
~
1.
we would
is a bounded sequence
1.
The construction of an a.e. convergent subsequence can now be
accomplished by the diagonal procedure as used, for example, in the
proof of the theorem of ReIly-Bray.
Equivalently, we can say that the
existence of such a subsequence, is a consequence of the criterion for
the relative compactness of subsets of
00
R
(see,
e~g.,
Billingsley
0
[1968, page 219]).
We now recall that, in Chapter I, we introduced the notation
p
for the metric corresponding to convergence in probability.
In any metric space
we write
d(s, A)
(S, d),
for
s
E
S
for the distance between
s
and
and
defined to be:
d(s, A)
=
inf
t
€
d(s, t).
A
A a subset of
A,
which is
S,
21
THEOREM 2.2.3.
Let
n
(n, A, p).
atomic probabiZity space
quence,
Suppose that the corresponding se-
of d.f. 's is weakZy convergent to a d.f.
{Fn },
G,
we denote by
be a sequence of r.v. 's defined in an
{X}
(n, A, P),
the famiZy of r.v. 's in
Then, if
F.
whose d.f. is
F,
we have:
G is not empty.
i)
iiJ
converges to zero as n
p(X,
G)
n
Proof·
Clearly, every subsequence of
to the same limit
F.
by Theorem 2.2.1,
{F}
n
exists a subsequence
that:
X(w)
= lim,
n
Hence,
of
X ,(w)
n
{F }
n
is weakly convergent
is weakly compact and therefore,
{F}
n
is tight.
{X.}
n
tends to infinity.
By the previous theorem, there
{X}
n
which converges a.e.
is a r.v. on
(n, A, P),
whose d.f. is
To prove iiJ let
{X ,}
be any subsequence of
n
corresponding sequence,
{F
n
,},
{X }.
n
converges a.e. and hence, in probability to a r.v.
n
G)
~
Since the
of d.f. 's is again tight it follows,
by the previous theorem, that we can find a subsequence,
p (X '"
F
G is not empty.
and that implies that
(2.2.7)
It follows
p (X '" Y)
='>
n
0
Y € G.
{Xn"}'
which
Hence:
as
This shows that any subsequence of the numerical sequence
{p(X ,G)},
n
contains a further subsequence which converges to zero.
The
result follows.
CoroZZary.
exists on
(0,
In the same conditions, as in the above theorem, there
A,
r.v.'s, with d.f.
bility, as
P)
F,
a sequence,
{Y},
such that
X - Y
n tends to infinity.
of identically distributed
n
n
n
converges to zero, in proba-
22
Proof.
that:
lim
Let
n-+oo
a
n
{an}
be a strictly positive numerical sequence, such
For each
= 0.
n?: 1,
there exists
Y
€
n
G such
that:
(2.2.8)
p (X ,Y )
n
n
<
Furthermore, it is clear from the definition of
p
p(X ,Y ) = p(X -Y ,0).
n n
n n
By the previous theorem and the assumption about
that:
{a }
n
it follows
that:
p(Xn -Yn ,0)
°
-+
as
n -+
00.
The result now follows from the equivalence between p-convergence and
convergence in probability.
Remarks.
1) In the first comment, that followed Theorem 1.1.3 of
Chapter I, we introduced the concept of an admissible probability measure for a probability space.
In the real line we will talk about ad-
missible d.f.'s and part i) of the previous theorem says that the class
of admissible d.f.'s, for an atomic space, is closed under weak convergence.
This result may seem contradictory, if we consider the fact
that there are sequences,
{F},
n
of purely discrete d.f.'s, which con-
verge weakly to a continuous d.f.
F.
Of course, the answer to this
apparent contradiction lies in the fact that no sequence of r.v.'s,
with those d.f. 's, can be defined in an atomic space.
2)
{X},
Consider, now, a sequence,
Bernoulli r.v.'s, with probability
n
p
of identically distributed
of success,
0 < p < 1.
Of
course, the corresponding sequence of d.f.'s is admissible for any probability space, which contains an event with probability
p.
However,
23
if we require the
xn 's
to be independent it follows, from the above
results and the central limit theorem, that such a sequence of r.v.'s
cannot be defined on an atomic space.
Since independence plays an im-
portant role, in the construction of such examples, it is probably
worthwhile to consider the problem of the existence of sequences of independent r.v.'s on atomic spaces.
Renyi [1970, pages 167-168] shows that there are atomic probability
spaces where, non-trivial, sequences of independent events exist but,
our freedom, to choose the values of their probabilities, is severely
restricted by the atomic structure.
In the theorem below, we rephrase
Renyi's result in a way which is more convenient for our purposes.
THEOREM 2.2.4.
events, on an atomic
n
~
(Renyi)
Let
p~babiZity
be a sequence of independent
{B}
n
space
(n, A,
P).
Let
q
n
... P(B ),
n
Then, we have:
1.
(Xl
(2.2.9)
min(q , 1-q ) <
n
n
(Xl.
It is well-known that, in atomic spaces, convergence in probability
and convergence a.e. are equivalent (see, e.g., Neveu [1965, page 48]).
We now use Renyi's result to show that, with the assumption of independence, convergence in law is also equivalent to the other two.
THEOREM 2.2.5.
Let
{X}
n
be a sequence of independent r.v.'s de-
fined on an atomic probabiZity spaae.
If the aorresponding sequenae of
d.f. 's is weakZy aonvergent to a d.f.
F,
Xn's
converge a.e.
F is degenerate and the
24
Proof.
Let
x
denote a continuity point of
=
B
n
{w: X (w)
n
form a sequence of independent events.
F.
The events:
s x}
Put:
q
n
= P(Bn ),
all
n;:: 1.
From the assumption of weak convergence, we have:
(2.2.10)
lim
Cln = q,
Osq:sl.
From (2.2.9) it follows that:
(2.2.11)
lim
n -+
00
min[q , (l-q )]
n
n
= O.
But,
all
n ;:: 1.
From (2.2.10) and (2.2.ll), we have:
1 - 12q-ll
It follows that,
degenerate.
q
= O.
is either zero or one, which shows that
F
is
Since convergence in law, to a degenerate d.f., implies
convergence in probability and the latter is, in atomic spaces, equivalent to convergence a.e. the proof is complete.
o
2.3. The general case.
In this section we discuss the problem in general probability
spaces, that is, spaces which have both an atomic and a non-atomic part.
If, as in Definition 2.1.3, we denote by
of
(n, A, P)
(2.3.1)
we have:
A the union of all the atoms
O
25
In order to reduce the problem to the situations discussed in the
previous sections, we introduce two new probability measures,
P2'
(n, A).
on the measurable space
p(BnA
(2.3.2)
P (B)
1
=
P(B/A )
O
=
(2.3.3)
P (B)
2
=
P(B/A~)
=
and
P
Because of (2.3.1),
Pi
For all
B
E
A
P
1
and
define:
o)
P(A )
O
p(BnA~)
P(A~)
are both well defined and they are,
2
(n, A).
clearly, probability measures on
checked that the probability spaces,
Furthermore, it can be easily
(n, A, P )
(n, A, P2 )
and
1
are
atomic and non-atomic respectively.
Some remarks on notation.
Since we now have three different proba-
bility measures in the same space, we will, whenever confusion is possible, state between parenthesis, the measure under consideration at
that moment.
So far, we have considered r.v.'s as being defined in
probability spaces, since we do not distinguish between two real valued
measurable functions defined on
P-measure zero.
which differ only on a set of
In this section, each real valued, measurable function
(n, A)
defined on
(n, A),
determines three different r.v. 's.
that any element of the P-equivalence class of
(n, A, P1 )
other corresponding r.v. 's in
P
1
and
P
2
are
P,
X with respect to
(n, A, P 2 )
and
Finally, if
we will write
P
1
and
P
2
F(1)
F
and
P.
since, both
The con-
denotes the d.f. of
F(2)
respectively.
It follows, from (2.3.2) and (2.3.3), that:
(2.3.4)
determines the two
absolutely continuous with respect to
verse is, of course, not true.
with respect to
X,
Observe also,
X
for the d.f.'s of
26
The above remarks suggest that a way to deal with questions, concerning
r.v.'s in a general probability space, is to try to solve the problem,
separately, in the associated atomic and non-atomic spaces.
If solu-
tions can be found, in both cases, we hope that a suitable combination
of them will provide an answer for the original question.
In our problem, however, this approach will not allow us to apply
directly the theorems of the previous sections.
This is so because, as
does not
the following example shows, the weak convergence of the
{F(l)}
necessarily imply the weak convergence of
Example 2.3.1.
Let
(~,l]
A is clearly a sub a-field of
atom is
m to
(~,l].
Since the
X 's
n
weakly convergent.
{F(2)(x)}
n
(n, A,
Define, for all
it follows that the
and
A,
X2n (w)
=
X _ (w)
2n 1
=
0-
A be the
and all Borel subsets of
is a probability space whose unique
P)
n
~
{O
1
(
1:
if
W E:
(.~,
if
w E:
[O,~]
if
w e:
[O,~J
if
w€
(~ ,1]
P,
1]
identically distributed
form a constant sequence and are, therefore,
However, for each
x,
°
< x < 1
both
are oscillating sequences of zeros and ones, which
shows that neither one of the sequences,
verges weakly.
Let
B the
B and if P denotes the re-
are, with respect to
F 's
n
[0,1],
m be the Lebesgue measure.
class of sets, formed by the interval
striction of
and
n be the closed interval
field of its Borel sets and
[O,~].
n
con-
27
The next two lemmas will show how to overcome this difficulty.
The
first one shows that, although weak convergence may be lost in the decomposition of the space, tightness is preserved.
Lemma 2.3.1.
(Q,
space
A, P).
d.f.'s is tight.
Proof·
Let
{X}
n
be a sequence of r.v. 's on the probability
Then, both
Let
tight, there exists a closed, finite interval
P({w: Xn(w)
€
[a,b]})
Clearly, for any 2 sets,
(2.3.6)
P[BnC]
~
Since
B
€
[a,b],
1 - eP(A )
>
of
the other case being
arbitrary, be given.
e > 0,
n
are tight.
and
We prove the result for
totally analogous.
(2.3.5)
{F},
Suppose that the corresponding sequence,
o
A and C
€
{F}
n
is
such that:
all
1.
~
n
A, we have:
P[B] + P[C] - 1.
Hence, it follows that:
=
Hence, by (2.3.2):
all
n
~
1.
The result follows.
Observations.
1. We choose to present a direct proof of the lemma
above because it is simple and depends only on the definitions.
shorter proof can be obtained from the fact that both
absolutely continuous with respect to
P.
P1
and
A
P2
are
28
2. Note that the tightness of
dent for
{F}
n
are tight,
{F}
n
Lemma 2.3.2.
(n, A, P).
space
quences
to be tight.
or
is not suffi-
However, if both,
is tight.
{X}
Let
be a sequence of r.v. 's in the probability
n
Then, the weak convergence of any two of the se-
{F} {F{l)} {F(2)}
n'
n
'
n
imply the weak convergence of the third.
Furthermore, if the weak-limits are, respectively,
and
F,
F(2),
(2.3.4) is true.
Proof.
It is enough to show the result for one pair of sequences,
the proof for the other two being totally analogous.
is weakly convergent to
F
Assume that
is weakly convergent to
and
From (2.3.4) we have, for all real
x
and all
n
~
F
n
F{l).
1:
(2.3.7)
Let
D
1
denote the set of points in the real line, which are con-
tinuity points of both
lows that
D
1
F
and
Since
is countable, it fol-
is dense in the line.
{F (2)}
It follows from our assumptions and from (2.3.7) that
verges at every point of
it follows that
Since
is dense and
is weakly convergent to a d.f.
now implies the validity of (2.3.4) for all real
uity point of the three d.f. 's
such
XIS
F, F{l)
F(2).
{F(2)}
n
F(2).
con-
is tight
(2.3.7)
which is a continSince the set of
is again dense in the line (it has a countable complement),
the validity of (2.3.4), for all real
complete.
and
x
n
x, follows and the proof is
29
We are now in a position to extend the results of the previous
sections to general probability spaces.
THEOREM 2.3.1.
Let
{X}
n
(n, A, P)
probabiLity space
be a sequence of r.v.
and suppose that the sequence
d.f. 's is weakLy convergent to a d.f.
(n, A, P),
class of r.v. 's on
Then~
F.
iiJ
p(X, G)
n
Proof.
{F(1) }
n
~
0
as
Since
{F }
and
{F(2)}
n
n
we have:
~ ~.
is tight it follows, from Lemma 2.3.1, that
are tight.
n
{F(i)}
vergent subsequence of
n
and let
{F(1)}
Let
is weakly convergent to a d.f.
be its limit.
G
all
A, P1 ),
exists in
{X ,},
n
It follows that
x
€
1
R .
considered as a sequence of r.v. 's in
satisfies the assumptions of Theorem 2.2.3.
(n, A, Pi)
other hand, since
whose d.f.
(P )
Z
a r.v.
(n, A, P )
2
is
The sequence
H and we have:
(2.3.8)
The sequence
denote a weakly con-
n'
satisfies the conditions of Lemma 2.3.2.
(Q,
of
if G denotes the
F~
whose d.f. is
{F}
n
G is not empty
iJ
both
defined in the
's~
Y whose d.f.
(Pi)
is
Hence, there
G.
On the
is non-atomic we can choose a r.v.
Z
H.
Define:
(2.3.9)
(I
A
o
= Y(w)IA (w) + Z(w)I (w)
o
AO
indicator of the set Ao)'
X(w)
is the
(2.3.2) and (2.3.3), plus some easy calculations, imply that the
d.f.
(P)
of
implies that
X is equal to the right-hand side of (2.3.8) and that
G is not empty.
30
To prove ii) we will show, as in the proof of Theorem 2.2.3, that
any subsequence of the numerical sequence,
ther subsequence which converges to zero.
{X }.
quence of
Since
n
contains a fur-
n
Let
{X ,}
n
be any subse-
is a tight sequence of d.f.'s it follows
{F ,}
n
{F(l)}
by Lemma 2.3.1 that both
{p (X ,G)},
and
n'
{F(2)}
are tight.
n'
Choose a sub-
{F (1) } which is weakly convergent to a d.f. G. By Lemma
n"
{F(2) } is also weakly convergent and we let H denote its
n"
sequence
2.3.2,
{X ,,} ,
limit.
considered as a sequence of r.v.'s in
n
sat-
(S1, A, Pi)'
isfies the assumptions of Theorem 2.2.3 and, considered as a sequence
in
A,
(S1,
P ),
2
it satisfies the assumptions of Theorem 2.1.3.
corollary to Theorem 2.1.3 there exists, on
{Zn"}'
A, Pi)' a sequence,
(S1,
of identically distributed r.v. 's, with d.£.
(2.3.10)
X"-Z,,
n
n
Prob
--;:.
Similarly, there exists on
0
(S1,
X " - U"
n
n
Prob
--;:.
such that:
G,
[Pi]'
A, P2 ) a sequence {Un"} of identi-
cally distributed r.v. 's, with d.f.
(2.3.11)
By the
H,
such that:
0
[P ].
2
::
Z "IA + U "I
n O n A~
Define, as in (2.3.9):
y"
n
Clearly, the
X " - Y"
n
n
y ,,' s
n
have d.f.
F
and it remains to be shown, that
converges to zero, in probability
(P).
Let
E:
>
0,
arbi-
trary, be given.
{w:
Ixn ,,-Yn "I
n
c
AO'
>
d
= ({w:
Ixn ,,-Yn "I>
d n
A )
O
U
{w:
Ixn u-Yn "I>
d)
31
Hence
P ({w:
Ixn lJ-Yn "I
>
IXn ll-Zn "I
d) = P ({ w:
+ P({w:
IXn ,,-un II I
+ P(A~)
• P 2 ({w:
> e:} n
> e:} n A )
O
A~) = P(AO) • Pi ({w:
IXn"-Zn" I
> d)
Ixn"-u n " I > e:}).
By (2.3.10) and (2.3.11) both terms in the last sum·converge to
zero as
n"
+
00.
Hence, it follows that:
p(Xn "' Yn II)
+
0
as
n +
00
[P].
But, we also have:
nil
all
~
1.
From this we have:
p (X ,,, G) + 0
n
as
n
+
00.
The result follows.
Remarks.
1. The corollary to Theorem 2.2.3 is also valid here and
the proof is similar.
2. The observations that followed Theorem 2.2.3 also apply in the
present situation since, the existence of a unique atom is enough to
prevent the admissibility of continuous d.f.'s.
In particular, the
proof of Renyi's result follows without any changes and in the proof of
Theorem 2.2.5, we substitute the equivalence between the two types of
convergence (which is not true here), by a combination of (2.2.9) with
the Borel-Cantelli lemma.
CHAPTER III
GENERALIZATION TO COMPLETE, SEPARABLE METRIC SPACES
In this chapter we are going to consider sequences of random elements, defined on a probability space
(S, S),
where
S
(n, A, P)
and taking values on
is a complete, separable, metric space and
a-field of its Borel sets.
S is the
As mentioned in Chapter I, we will make ex-
tensive use of the class of partitions of
in the proof of his result (Theorem 3.2.2).
S
introduced by Skorokhod
The main part of Section
3.1 will be devoted to the study of the properties of this class of
partitions.
3.1. The Skorokhod partition of S
Definition 3.1.1.
Let
probability measure on
(S,
Q({s})
=0
for all
s
€
S be a separable metric space and
S).
Q a
Q is said to be non-atomic if
s.
Since we have been using the concept of non-atomic in a more general context (with respect to an abstract measurable space
(Q, A»,
we
decided to include here the proof of the equivalence of the two definitions on separable metric spaces.
33
THEOREM 3.1.1.
abiZity measure on
for
(S, S,
is that
Q)
Q({s})
Proof:
space.
= 0 for aZZ
So
Then~
s
S.
E
(S, S, Q)
s
E
S.
S.
is a non-atomic probability
Since the only subsets of
itself and the empty set it follows, from Definition
(S, S, Q)
is an atom of
{sO}
i f and only if
Q({so}) >
o.
is assumed to be atom-free it follows that
Q to be such that
Conversely let us now assume
all
a necessary and sufficient condition
denote an element of
(S, S, Q)
Since
S).
Q a prob-
to be a non-atomic probabiZity space (Definition 2.1.3)
{sO}
2.1.2, that
(S,
S be a separabZe metric space and
Suppose first that
Let
are
{sO}
Let
Let
B be an atom of
(S, S, Q).
assumptions lead to a contradiction.
Q({s})
= 0,
for
We will show that these
We are going to make use of the
fact that a separable metric space can, for each
n
~
1, be covered by a
countable sequence of disjoint Borel sets, each one with diameter (diam)
less than
lin
(see Appendix).
For each
n
~
1,
let
{An}
i
collection of Borel sets satisfying the above requirements.
denote the
Therefore,
we have:
oc
(3.1.1)
Since
B
=
U (A~nB)
~
i=l
for all
n
~
1.
B was assumed to be an atom, there exists, for each
a unique index in such that:
n
~
1,
34
Therefore we have:
(3.1.4)
Q(A)
n
Q(B)
=
and
Q(B-A)
n
= O.
From (3.1.4) it follows that:
(3.1.5)
QI
~ (B-An~
=
J
~=l
O.
But, we also have:
00
(3.1.6)
n
n=l
A
n
=
t [n~,<B-An)II
From (3.1.5) and (3.1.6) we conclude:
(3.1.7)
Q[n A]
n=l ~
Let now x
(3.1.8)
d(x,y)
and
:s;
Q(B).
=
y
denote 2 points in
diam[n A]
n=l ~
:s;
diam An
Since (3.1.8) is true for all
~
n
:s;
1,
n°On= 1An • We can write:
diam An
i
< lIn.
n
it follows that
d(x,y)
=0
n°O A
is either empty or a singleton.
n=l n
On the other hand the assumption that B is an atom implies Q(B) > 0
and hence we can conclude that
and hence it follows from (3.1.7) that there exists
So
€
S
such that:
00
(3.1.9)
Hence, from (3.1.7):
Q({so})
=
Q(B) > O.
The last statement contradicts the assumption that,
S €
S,
and the proof of the theorem is complete.
Q({s})
=0
for all
o
35
Definition 3.1.2.
A partition of a set
X is any finite, or de-
numerably infinite, collection of disjoint subsets of
X.
is
Given two partitions,
say that
Definition 3.1.3.
space
S,
S,
k
~
Definition 3.1.4.
Skorokhod partition
1
if each
J
J
P of a separable metric
A Skoroknod partition
1
P , P ,
l
of parti-
2
the elements of
(1/2)k.l
Let
P of
Pk
Pk are nonempty Borel sets
the elements of
whose diameters are at most
~
{B. }
each partition being a refinement of the preceding one and
such that for all
k
X we
B.•
is an ordered countable collection
tions of
whose union
of the same set
J
is a refinement (or a sub-partition) of
{Ai}
is a subset of some
Ai
{B.}
and
{Ai}
X,
P
S
be a probability measure on
(S, S).
is said to be P-aontinuous if for all
are P-continuity sets.
We will show in the Appendix that for each probability measure
(S, S)
on
there exists a P-continuous Skorokhod partition of
Remarks on notation.
used by Skorokhod [1956].
{Si : i 1
~ l}.
The elements of
For each fixed value of
{S i , ••. ,i _ ,i :
P2
i
k l k
sets of the element
1
1
since
i1
which are subsets of
S.
Pl
will be denoted by
{Si ,i : i 2 ~ I}
1 2
Si' In general,
denotes
1
k
~
l}
denotes the elements of
Pk
which are sub-
Hence, for all
k
>
1
and
we have:
all elements
(3.1.10)
P
We will adopt the same nested-type notation
1
the elements of
A
S
i 1 , .. "
i'
k-l
The name "Skorokhod partition" involves a slight abuse of language
P is actually a family of partitions of S.
36
Furthermore, since
(3.1.11)
=
P1
is a partition of
S it follows that:
s.
Observe that since each set
S.
~1'
i
is required to be non-
••• , k
empty we may not assume without loss of generality that the range of
is infinite.
Since
S
i1""'~_l
tains at least the number one.
is nonempty the range of
~
~
con-
Pm
With this notation we have a one-to-one correspondence between
and a nonempty subset of the Cartesian product
the set of natural numbers.
dicate that a
write
a
<b
;II
precedes
b
and
b
m
N ,
of
a ~ b
we write
m
N •
on the lexicographic order of
to indicate that
N denotes
Nm is a totally ordered set.
graphically and with this order
a
where
Nm can be ordered lexico-
Recall that
Given any two elements
m
N ,
a
to inWe will
precedes and is not equal to
b.
Recall that a totally ordered set is said to be well-ordered if
each nonempty subset has a first element.
well-ordered if for each nonempty subset
isfying
b
o
~ b
for each
b
€
B.
In other words,
B
there exists
(A, {)
b
o
€
is
B sat-
It is clear from the definition that
the induced order on a subset of a well-ordered set is a well-ordering
of that subset.
The following two lemmas are stated here for future
reference.
Lemma 3.1.1.
Cartesian product
Let
A
1
x
A and A be two well-ordered sets. The
2
1
A , ordered lexicographically, is a well2
ordered set.
Proof:
We give only a brief outline of the proof.
nonempty subset of
A x A .
2
1
Consider the set:
Let
D be a
37
=
U
a EA
2
Since
Let
{al~Al: (a ,a ) ~ D}
1
D is nonempty it follows that
a
2
2
B
1
is a nonempty subset of
A .
1
o be the minimum element of B , which exists since A is well
1
1
1
ordered.
Consider now the set:
=
B
2
is a nonempty subset of
It is now easy to show that
in the lexicographic order of
Lemma 3.1.2.
For each
A
2
and hence it has a minimum element
o
0
(a ,a )
1 2
is the minimum element of
D
A x A •
2
1
m ~ 1,
Nm ordered lexicographically, is a
well-ordered set.
Proof:
m=1
The result is trivially true for
well-ordered set.
since
N is a
The result follows by induction using the previous
lemma.
Definition 3.1.5.
Let
a probability measure on
P be a Skorokhod partition of Sand P
(S, S).
For a fixed
s
P
let
S
denote the unique element of
kl, ••• ,km
Define:
1f
0
(3.1.12)
F (s)
1
=
L
1
G (s)
1
=
= 1
1
k -1
1
i =1
(3.1.13)
k
m
P(Si )
1
F (s) + P(Sk ).
1
1
if
k
1
>
1
E
S
and each
which contains
m ~ 1,
s.
38
In general, for
m > 1,
define:
F _(s)
m-l
(3.1.14)
(3.1.15)
F (s)
m
if
I
F _ (s) +
m1
=
Observations.
m
= 1
k -1
m
=
G (s)
m
k
F (s)
m
1)
+ p(S
k ,k , ••• ,k
1
G (s),
P(s k ,k ,· .. ,k _ ,i )
1 2
m1 m
i =1
m
2
for
s
).
m
Sk
E
probability of the union of all elements
(i 1 ,···,i m) ~ (k 1 ,···,km).
Similarly
Si
is equal to the
k '
m
1"'"
m
km > 1
if
i
Fm(S),
for
P
of
l ' ... , m
s
E
for which
m
Sk
k'
is
l' ... , m
equal to the probability of the union of all elements 5
of
.
~
i 1 ,···,i m
Pm satisfying: (1 1 , .•. ,i ) ~ (k , •.. ,k ). In symbols we have:
m
1
m
(3.1.16)
Fm(s)
=
p[('
1
(3.1.17)
Gm(s)
p[('
=
1
2)
Fm and
iU)~(k
1 ' •.. , m
iU)J(k
1 , ... , m
1
k )5 i1 , ... ,i]
m
l' .. "
s,
E
the numerical sequences
{F (s)}
m
(5, S)
and
s
(3.1.18)
F(s)
3)
E
S,
{G (s)}
m
lim F (s)
m+ oo m
=
ural way a total order on
S.
G(5)
Hence,
In fact, if
s1
and
and hence for some
S,
if the set containing
lim G (s).
m
m+ oo
=
P
lexicographically \-1e induce in a natm
are in different elements of
s2
are
their limits exist and we put:
Observe that if we order
tinct points of
[0,1]
to
both monotone, the first increasing and the latter decreasing.
for each
Sk , .. ,k
m
1
k )Si , ... ,i]
1
m
m
Gm are both measurable functions from
and for fixed
s
l ' ... , m
~
s1
Pm,
are tl-10 dis-
s2
m
~
1,
precedes
and we may say that
precedes the set containing
and
52'
With
39
respect to this order the functions
F
and
G behave somewhat like
distribution functions associated with the probability measure
Lemma 3.1.3.
Let
partition of
Sand
functions
F
and
(3.1.19)
G(s)
Hence, if
P
Proof:
for all
m
=
G,
a probability measure on
F(s) + P({s})
is non-atomic
1
and
(5,
S).
s
S.
Then, the
satisfy the relation:
From (3.1.15)
~
P a Skorokhod
S be a separable metric space,
P
P.
s
the sequence of sets
€
{S
Sk
all
= G(s)
F(s)
S.
for all
€
we have that:
k'
1" •. , m
k ' ... ,k
i
S €
}
As
m increases, for fixed
s,
is monotonically decreasing, the se-
m
quence of its diameters converges to zero and
Sk , ... ,k ' all
i
m
By the same reasoning used in the proof of Theorem 3.1.1 (see
m~1.
(3.1.8»
s
€
we have that:
{s}.
(3.1. 20)
Therefore, we have:
(3.1.21)
=
III
(3.1.19) follows from (3.1.18) and (3.1.21).
P({s}) •
The last part of the lemma
follows from Theorem 3.1.1.
Remark.
tween
F
and
So far, it may appear that we have a perfect analogy beG and distribution functions on
i
R .
However, it should
be kept in mind that the properties of distribution functions, which
40
depend on the relations between the order and the topology of
no straightfonvard counterparts in the present situation.
P
though
depends on the topology of
P
with which the elements of
S,
In fact, al-
are chosen prevents the existence of a
S,
P,
induced by
and the
S.
Lerrma 3.1.4.
Let
{P }
n
be a sequence of probability measures on
is weakly convergent to a probability
{p }
and assume that
(S, S)
have
the relative arbitrariness
simple relationship between the order on
topology of
1
R ,
n
measure
P
on
(S, S).
tion of
S
and, for each
P denote a P-continuous Skorokhod parti-
Let
n
~
n
let
1,
m
{F :
m
~ l}
be the sequences of measurable functions associated with the pair
{G}
m
~ssociated
will denote the functions
and all
s
€
Proof:
(P,P).
with
and
{F }
m
For each
m
~
S:
lim
m=1
Similarly
in the way described in Definition 3.1.5.
(P , P)
n
Fn(s)
m
= Fm(s)
lim
and
n
-+
= Gm(s).
Gn(s)
m
00
m.
We will prove the result by induction on
and let
there exists
n
1
~
1,
S
n
such that
From (3.1.11) it follows that
S.
be any element of
s
€
1
Pi
and
s
€
S
n
If
1
we have by (3.1.12) , that
all
(3.1.22)
n
~
1
and also that:
F (S)
1
Suppose now that
n
1
>
1.
Consider
= O.
Again by (3.1.12) we have that:
n
1
= 1,
1
41
F(n)(s)
1
n -1
1
2
=
P (S. )
n ~1
i =1
n
all
~
1
1
(3.1.23)
n 1 -1
F (s)
1
I
=
P(Si ).
i =1
1
1
Since the
S
's
i
are P-continuity sets and
it follows that for each
lim
n+oo
i
= P(Si)
1
P (S. )
n
P
~1
1
,
1
~
i
1
~
n -l,
m = k.
which contains
s.
P
we have:
1
and, the result follows.
Assume now that the result is true for
the result for
converges weakly to
n
1
Let
S
n 1 ,n 2 ,··· ,nk
By (3.1.14) we have:
Fn _ (s)
k 1
m = k-l
and let us prove
denote the element of
Pk
if
n -1
k
=
+
L
~>1.
i =1
k
Similarly:
if
=
n
k
=1
n -1
k
F _ (s) +
k 1
Clearly, i f
~
=1
i
k
L
=1
the result follows by the induction hypothesis.
Otherwise:
n
=
lim
n -+
co
n
k
-1
2
F _ (s) +
lim
k 1
i =l n -+ 00
k
The result now follows by the induction hypothesis and the fact that the
42
elements of Pk are P-continuity sets. The proof of the result for
n
{G } follows from the result for {Fn } and (3.1.15).
m
m
o
We would like to point out that our objective is to use the
Skorokhod partition as a device that would allow us to adapt, for metric
spaces, the proofs given in Chapter II for the real line.
With this ob-
jectlve we proceed to find the probability distributions of the r.v. 's
F(s)
and
G(s)
Let
tion of
P
defined on
(S, S, P)
be a probability measure on
S.
Let
{F}
and
m
{G}
correspondence to each
Both
Bm
= {s:
B
and
m
t
Fm(s)
€
(5, S),
[0,1].
a Skorokhod parti-
P
be the two sequences of measurable
m
functions associated with the pair
(3.1.24)
with values in
[0,1]
as in Definition 3.1.5.
(P, P)
In
define:
A
< t}
m
= {s:
G (s)
m
Am are clearly dependent on
~
t}
t
m~l.
but, since
twill
be held fixed throughout the proofs, we do not indicate this dependence
in the notation.
In the following lemmas we establish some important properties of
the sequences of sets introduced in (3.1.24).
[0,1]
arbitrary but fixed real number in
and
In all of them
{F}, {G}
m
m
t
is an
have the
meaning given to them in Definition 3.1.5.
Lemma 3.1.5.
each element
whenever
1
all elements
ii)
element of
P
s, the entire set of
S
k
m~ 1
For each
i)
' ••• , k
Si
m
P.
m
is contained in
of
P
for each
m
i
l' ... , m
B - A,
m
m
m
m
~
Am contain, with
which contains s. Furthermore,
both
B
m
or
B and
m
Am ,
the same is true for
satis fying:
1,
is either empty or equal to an
43
Proof:
The first part of i) is an immediate consequence of the
fact that both
F
and
m
G are constant on each element of
m
second part comes from the fact that, for fixed
F
m,
increasing with respect to the lexicographic order on
To prove ii) observe first that if
contains, with each element
s.
tains
S
i
1
and
, ••• ,i
S.
~1
m
S
,
' •••
contained
k , ••• ,k
1
m
(k ,···,k).
m
1
i
~
.
'~m
and
s2
€
Bm - A.
m
in
are
m
is not empty it also
Pm which con-
the entire element of
s,
Pm,
Assume that
Pm are not empty we can take
Since the elements of
€
m
G
Pm.
Suppose that we have two distinct elements of
m
(i , .. · , i )
1
m
sl
B - A
and
m
P.
The
m
From (3.1.16) and (3.1.17) it
Sk l' ... , km
follows that:
(3.1.25)
On the other hand, since
s l ' Am
it follows from (3.1.24) that:
(3.1.26)
(3.1.25) and (3.1.26) imply that
we have that
s2
€
Bm-Am.
ments of
s2
t
F (s2) > t
m
and hence, by
B which clearly contradicts our assumption that
m
B - A
This shows that
m
m
cannot contain two distinct e1e-
Pm and hence the desired conclusion follows.
Lemma 3.1.6.
m
i) For each
G (s)
m
ii) For each
(3.1.2~
m
~
Gm(s)
1
and
= PCB m)
s
€
1
~
$
and
s
€
.
B
m
PCB ).
m
Bm-Am:
and
F (s)
m
= peAm).
o
44
Proof:
tains
s.
S
denote the element of P which conm
n 1 ,· .. ,n
m
By part i) of the previous lemma we have that:
i) Let
U
S.
~1"
(i 1 ,··· ,im)~ (n 1 ,·· • ,nm)
••
B .
m
c
.
'~m
Hence it follows that:
P(B).
~
m
The result now follows from (3.1.17).
ii) Since we assumed that
previous lemma
B - A is not empty by part ii) of the
m
m
p • By
is equal to an element
B - A
m
m
m
part i) of the same lemma, it follows that:
Bm
=
Am
= u
S
i 1 ,· •{ ,im
(i , · · , im;lt
) (n ,··,n)
m
1
1
The conclusion now follows from (3.1.16) and (3.1.17).
Lemma 3.1.7.
Proof:
PCB m) < 1.
i)
PCB )
m
~
t.
ii)
peA )
~
t.
i) If
m
= 1,
PCB )
m
we are done since
Hence there are elements of
Pm
~
t
1.
Assume then
which are not contained in
B and therefore the set:
m
L
= {(i 1 , ••• ,i
is a nonempty subset of
m
N .
m
1"'"
i
C
c B }
m
m
It follows by Lemma 3.1.2 that
minimum element that we denote by
3.1.5 we have that:
): Si
(n 1 , ••• ,n ).
m
L has a
By part i) of Lemma
45
(3.1.27)
Let now
=
Bm
s
U
Sn , ... , n .
1
m
€
=
Fm(s)
0.1.28)
But
Si'
J
(k '···' i ) 1 ( n1,···,n )
m;Z:
m
1
l' ••• '].m
By (3.1.16) and (3.1.27) we have that:
P(B).
m
siB m and hence by (3.1.24):
(3.1. 29)
~
F (s)
m
t-
(3.1.28) and (3.1.29) imply
P(B)
m
~
t
and the proof of i) is
complete.
R = {t
ii) Let
for
t
of
P,
{s: G (s) = t}
m
Pm.
m
Observe again that, since
R.
€
[0,1]: P({s:G (s)=t})
€
(n , ... ,n )
1
m
{(i , ... ,i ): Si
l
m
{s:G(s)=t}
m
and
G
inition of
m
G is constant on the elements
m
be the first element of the set
..
1, .••
First we show ii)
is a finite or countable union of elements of
m
Let
o}.
>
c
{s:G (s)=t}}.
Assume now that
m
m
(n , ... ,n)"/:(k , ... ,k).
m
1
m
1
,1
that
P(S
) =
k , •.. ,k
m
1
o.
s
k l ,···,k m
c
It follows from the def-
Hence, by part i) of Lemma
3.1.5, we have that:
P(A )
(3.1. 30)
=
m
From (3.1.17), it follows that, if
=
P(A)
m
Consider now
{t},
n
t
n
€
R,
t
t
n
€
R,
< t,
t i R.
all
n
~
S
€
=
G (s)
m
S
:
n , ... ,n
1
m
t.
Assume first that we have a sequence
1,
lim
~
n~
t
=
n
<Xl
(3.1.31)
{s: G (s) <
m
d
=
U {s: G (s) :s;
m
n=l
t
n
}.
t.
Then, we can write:
46
t t R,
Hence, since
(3.1. 32)
we have:
=
P(A)
m
=
P({s: G (s) < t})
m
Therefore, since
tn
~ tn})n~l
({s: Gm(s)
~
P(U{s: G (s)
t }).
n
is an increasing sequence
of sets:
=
peA )
(3.1. 33)
m
lim P({s: G (5)
m
~
=
t })
n
The case where there exists a sequence
can be handled in a similar way.
t
R,
E
=
peA )
m
E
R,
t* < t
t i
n
R.
>
to
tall
m
Otherwise since
~
P({s: G (s)
m
t*
E
=
t})
If there exists no
P({s: G (s) ~ t})
we would have that
there exists a largest element
=
t
{t },
n
n =
n
1,
~
to
and we would be done.
P(A)
m
t
Hence, we have proved that for all
Finally, consider the case
t*
lim
R,
t*,
= peAm) =
°
R is a closed subset of
t* < t.
~
t
[0,1]
Clearly:
~
P({s: G (s)
m
=
t*})
t* < t.
o
This completes the proof of the lemma.
THEOREM 3.1.2.
Let P be a probahiZity measure on
a Skorokhod partition of S.
As
before~
and
{F}
m
are the funotions introduoed in Definition 3.1.5.
eaoh
t
E
[0,1]
let
Bm and
values of
t
(3.1.33)
{s: F(s) < t}
E
[0,1]
{G}~
m
Por
Am be given by (3.1.24).
lim~
for whioh
P(B m)
= t:
00
c
nBc
{s: G(s)
~
t}.
m
m=l
Henoe, for all suoh values of t:
(3.1.34)
P({s: F(s)
<
t})
al~
~
t
~
P({s: G(s)
~ t}).
(S, S)
and P
F and
m
~
Then~
1
G
and
for all
47
Proof:
t
E
[0,1].
The first inclusion in (3.1.33) is in fact true for all
It follows from (3.1.24) and the fact that
{Fm(s)}~l
is
increasing that:
(3.1.35)
{s: F(s)
< t}
c
{s: F (s) < t}
m
= Bm
all
m
~
1.
The first inclusion in (3.1.33) follows from (3.1.35).
To prove the other inclusion, let
By part i) of
Lemma 3.1.6, we have that:
(3.1. 36)
G (so) :;; P(B )
rn
m
all
m
~
1.
Therefore, by (3.1.18) we have that:
:;;
Hence,
So E
{s: G(s) :;; t}
=
lim P(B)
m
t.
and the proof of (3.1.33) is complete.
(3.1.34) now follows from the fact that
{B}
m
is a decreasing se-
quence of sets and hence
=
lim P(B ).
m
For the next theorem we assume that we are given a non-atomic probability measure on
(S, S).
From Theorem 3.1.1 it follows that for the
existence of a non-atomic measure on
be an uncountable set.
(8, S)
it is necessary that
This condition is also sufficient if
8
S
is
separable and complete (see, e.g., Parthasarathy [1967, pages 53-55]).
THEOREM 3.1.3.
(S, S)
G(s),
Let P be a non-atomic probability measure on
and P a Skorokhod partition of
defined on
(8, S, P)
s. Then, the r.v.
is uniforrnZy distributed on
F(s)
[0,1].
=
48
Proof:
fact suppose that
(3.1.37)
m ~ 1,
We first observe that for each
S
€
A
nBc.
m
It follows from (3.1.24) that:
m
and
G (s) :::; t
m
P,
m
It follows from (3.1.15) that the element of
s
In
belongs has P-measure zero.
to which
Therefore, since
not
empty is a finite or countable union of elements of
Hence, we have for all
(3.1. 38)
P(B -A)
m m
m
~
1:
= P(B) - P(A ).
m
m
Suppose now that for a given value of
(3.1. 39)
Pm it follows that
t,
we have:
lim P (B ) > t.
m
From (3.1.38), (3.1.39) and part ii) of Lemma 3.1.7 it follows
that:
(3.1.40)
P
[~
m=
(B -A )]
m
1
m
=
lim P«B -A »
m m
>
o.
However, by part ii) of Lemma 3.1.5 we know that
element of
Pm for each
of Theorem 3.1.1,
00
m
~
1.
nm=l (Bm-Am)
B
m
- A
m
is an
By the same argument used in the proof
is either empty or a singleton.
Hence,
(3.1.40) implies the existence of an element
So
This clearly contradicts the assumption that
P is non-atomic and there-
fore for all
t
€
[0,1]
we should have:
by the previous theorem and the fact that
t
€
[0,1]:
lim
~
€
S such that:
P(B) = t.
m
Therefore,
F = G we have for all
49
P({s: F(s)
t})
<
~
t
~
P({s: F(s) ~ t}).
o
The result follows.
3.2. The non-atomic case
We are now going to apply the techniques developed in the previous
section to extend to metric spaces the results of Chapter II.
Our first
theorem shows how to associate to each r.e. defined on a non-atomic
probability space a convenient uniform r.v.
It is, therefore, the
equivalent for metric spaces of Theorem 2.1.2 of Chapter II.
Let
THEOREM 3.2.1.
bility spaoe
du~ed
G
by
(Q,
X on
A,
X be a r.e. defined on a non-atomio proba-
and let
P)
(8, $).
x denote the probability measure inLet P be a Skorokhod partition of 8, F and
P
x ' P) as
Z be any uniformly dis-
be the two measurable funotions determined by the pair
desoribed in Definition 3.1.5.
tributed r.v. defined on
Finally, let
A, P).
(Q,
For
s
€
8 and
z
€
(P
[0,1],
defi~:
H(s,Z)
(3.2.1)
Then,
on
1
F(s) + P({w: Z(w) ~ z} n X- ({s}».
is a measurable funotion from
H(s,z)
([0,1], B)
=
and the r.v.
U(w)
= H(X(w),
z(w»
«s
x
[0,1], $ x
B»
to
is uniformZy distributed
[0,1].
Proof:
The proof that
serve only that
F(s)
H is measurable is straightforward.
Ob-
is measurable and that the second term is dif-
ferent from zero for, at most, a countable number of values of
s.
50
By Lemma 3.1.3 we can write:
(3.2.2)
=
G(s)
1
F(s) + P(X- ({S}».
=
From (3.2.1) (3.2.2) and the definition of
(3.2.3)
Let now
and
~
F(X(w»
°
t,
U(w)
~
~
t
~
1,
G(X(w»
all
be given.
D,
w
it follows that:
~
€
In correspondence to
t,
P
x
P consider the sequences {B}
and {A} introduced in (3.1.24).
m
m
We have two cases to consider:
1)
For the given value of
lim PX(B )
m
t,
= t.
It then follows
from Theorem 3.1.2 that:
(3.2.4)
P({w: F(X(w»
~
< t})
t
~
~
P({w: G(X(w»
t}).
On the other hand, from (3.2.3) we have that:
{w: U(w)
<
t}
c
{w: G(X(w»
~
t}
{w: F(X(w»
<
t}
~
t}.
(3.2.5)
{w: D(w)
c
(3.2.4) and (3.2.5) imply that for all values of
lim
~
(3.2.6)
2)
lim
~
PX(B)
m
=
t
>
t.
(3.2.8)
S
o
€
[0,1]
for which
~
t
P({w: D(w)
B - A
m
~
t
t}).
we have:
By the argument used in the proof of Theorem 3.1.3,
it follows that there exists
(3.2.7)
~
Assume now that for the given value of
m
E
we have that:
P({w: D(w) < t})
PX(B)
t
m
all
So
€
m2 1
S,
such that:
51
Hence, by part ii) of Lemma 3.1.6 we have that:
0.2.9)
Gm(so)
=
PX(B m)
Fm(sO)
=
PX(Am)·
Therefore, it follows from our assumption and from part ii) of
Lemma 3.1.7 that:
(3.2.10)
=
=
z
Consider now, for every
(3.2.11)
Clearly
fez)
fez)
(3.2.12)
=
P(x
-1
E
[0,1]:
({so}) n {w: Z(w)
is a continuous function of
f(O)
= 0;
~
z}).
z
and we have:
f(l)
On the other hand, from Lemma 3.1.3 and (3.2.10) we have that:
(3.2.13)
Since a continuous function defined on a compact set assumes all
the values between its minimum and its maximum it follows that there
exists
Zo
E
[0,1],
such that:
=
0.2.14)
We will now show that:
(3.2.15)
{w:
U(w)
<
tl
c
t-1[ilAm]u{x-1({SQI)n{w: Z(W)<'Qlj
c
{w: U(w) ~
d.
52
To prove the first inclusion in (3.2.15) let
U(w o)
<
t.
It follows from (3.2.3) that
F(X(w
»
O
o be such that
W
<
t.
Recall now
that, as mentioned in the proof of Theorem 3.1.2, the first inclusion in
(3.1.33) is valid for all
t
€
[0,1].
Hence, we have that:
00
n
(3.2.16)
B •
~l
m
Observe that we have either
X(w o)
€
B -A
m m
m ~ 1.
for all
X(w
o)
€
Am
for some
m~ 1
or
In the first case we are done and in the
latter it follows that we must have
X(w
o) = sO' Therefore, in this
last case we can write:
(3.2.17)
On the other hand from (3.2.1) and (3.2.11) it follows that:
(3.2.18)
=
Therefore, by (3.2.14) we have that:
(3.2.19)
Now if we observe that
tion of
z
H(s,z)
is, for fixed
s,
an increasing func-
it follows from (3.2.17) and (3.2.19) that
Z(w O)
<
z00
This completes the proof of the first inclusion in (3.2.15).
To prove the other half, let
wi
be such that:
We have two cases to
consider:
1)
lows from (3.1.24) that:
It fo1-
S3
Hence, since
{G (s)}
m
is a decreasing sequence:
G(X(w 1»
Therefore, by (3.2.3) we have that
001 E {w:
U(W ) Stand hence
1
U(w) S t}.
X(W 1)
2)
S to
= So
Z(w 1 ) ~ z00
and
Hence, from the definition of
U
we have that:
Hence, it follows from (3.2.19) that
U(W ) Stand the proof of
1
(3.2.15) is complete.
On the other hand, we have by (3.2.11):
Hence it follows from (3.2.10) and (3.2.14) that:
(3.2.20)
From (3.2.15) and (3.2.20) it follows that (3.2.6) is valid for all
t
E
[0,1]
and this implies that
U(w)
is uniformly distributed in
o
[0,1] •
Remarks.
1) If
P
x
is a non-atomic measure on
(5, S)
we have
that:
F(X(w»
= U(w)
=
G(X(w»
and hence the theorem above reduces to Theorem 3.1.3.
2) It is not necessary that the r.v.
buted.
Z(w)
be uniformly distri-
Any continuous r.v. taking values on a compact set of
be enough for our purposes.
R1
would
54
In Chapter II, we saw how the inverse function
Lebesgue measure.
role of
[0,1]
B denotes the Borel sets of
Here,
F
-1
-1
of ad. f.
([0,1], B, L)
determines a r.v. on the probability space
F.
F
and
L
F
whose d.f. is
is the ordinary
We now introduce the construction that will play the
in the present context.
The idea of this construction is
due to Skorokhod [1956] and we reproduce his proof in the theorem below
as a matter of convenience.
THEOREM 3.2.2. rSkorokhodJ.
Let
8
be a complete, separabZe
metric space and Q be a probabiUty measure on
exists on
([0,1], B, L)
biZity distribution is
Proof:
P
Let
a r.e.
with vaZues on
(8,
S) whose proba-
Q.
denote a Q-continuous Skorokhod partition of
In correspondence to each
[0,1],
X
Then there
(5, S).
S.
Pm we are going to associate a partition of
by means of intervals
~
i '
i 1 ,i 2 ,···, m
which we take to be 1eft-
closed and right-open, satisfying the following conditions:
~i'l ' ... , i'm
The length of
Q(S
From each element
Pm
2)
-
Xi1 ,i
,···,i ·
2
m
For each m
(3.2.21)
~
1 and
xm(w) =
xi , ... ,1.
1
Since each partition in
we have:
(3.2.22)
[0,1]
W E
if
m
P
if
).
i 1 ,···,i m
we choose a point
we define:
W £. ~.
1
1 , .. "
i
.
m
is a refinement of the preceding ones
55
It follows that
complete,
lim
{X (w)}
X (w)
m
~
X(w)
=
is
w € [0,1]:
lim
m ~
S
exists.
We define for all
(3.2.23)
is a Cauchy sequence and since
m
<Xl
X (w).
m
To complete the proof it remains to be shown that the probability
distribution of
and
sets.
LX-
1
X is
Q.
We claim that it is enough to show that
are equal on the Borel subsets of
A
which are Q-continuity
S
In fact, this class of sets is a field and it can be shown that
the a-field generated by this field is
Let
A denote a Q-continuity set.
all elements of
P
S.
denote the union of
Let
which are contained in
m
the union of all elements of
Pm
A and let
A,(m)
which are not contained in
denote
S - A.
Clearly:
(3.2.24)
A(m) cAe A' (m) .
Observe also that from the construction of
=
(3.2.25)
X we have:
Q(A(m».
Similarly we have that:
(3.2.26)
L({w: X(w)
Let now
boundary of
C(m)
A
€
A,(m)})
=
denote the set of points whose distance from the
is less than or equal to
(~)m.
Observe now that if
S.
. c A,(m) - A(m), S1
.
contains points of the boundary
1 1 ,···,1 m
1,···,1m
of A and hence the distance from a point in S.
.
to the boundJ.
ary i s I ess t h an
(3.2.27)
(~L)m.
Hence, we h ave t h a t :
1 ,···
,J. m
56
But,
{C(m)}
boundary of
A.
is a decreasing sequence of sets whose limit is the
Hence, since
A is a Q-continuity set, it follows
that:
(3.2.28)
o
The result follows.
Remark.
va1s
i
1:::..
[F (s) ,G (s»,
m
for
m
determined by the pair
Corollary.
Let
exists on
G
m
are
X,
Q a
Then, there exists in any non-atomic
with values in
(n, A,
Since
(n, A, P)
P)
S,
whose probability dis-
is non-atomic, by Theorem 2.1.1, there
a uniform r.v.
Z(w)
Z.
The construction of the above
produces the desired r.e.
Z be a uniformly distributed r.v., defined on
Let
(n,
A,
where
S
the non-atomic probability space
(S, S)
probability measure on
and let
P)
Q be a
is a complete separable metric
X denote the r.e. obtained by applying the Skorokhod con-
Z.
struction to the r.v.
(3.2.29)
and
Q.
Lemma 3.2.1.
Let
F
m
as in Definition 3.1.5.
(S, S).
theorem applied to each value
space.
where
.
1" .• '~m
S be a complete, separable metric space and
probability space a r.e.
Proof:
Si
S E
(Q, P)
probability measure on
tribution is
given by the inter-
coincides with the partition determined by the in-
~1' 2,· .. ,i m
tervals:
[0,1]
Observe that the partition of
P({w: X(w)
E
Then, for all
s.
~1
m
~
I
1 } n {w: Z(w) i
' .•. , m
and all choices of
1:::.
1 1 ' .• "
1})
m
= 0.
57
Proof:
The result is trivial and it is in fact assumed in the
proof of Skorokhod's result (see (3.2.25) and (3.2.26».
if
Z(w)
E
~k
l' .. "
X(W) E Sk , .•. ,k '
1
m
k '
m
(3.2.30)
Observe that
Hence, we have that:
c
Hence it follows that:
(3.2.31)
P({W: X(W)ES.
~1
=
Q(S.
:s;
Q(S
~1 ' . "
i
i , .. ,i
1
Z(w)t~.
i
}n{w:
m
)
n Sc
i 1 ,··,i m
' •• , m
n SC
i
m
~1
)
1
i })
' •• , m
O.
=
,··,i m
The last equality follows from the fact that
P
0
is Q-continuous.
We now prove a more general version of Theorem 2.1.3 of Chapter II.
THEOREM 3.2.3.
Let
n
non-atomic probability space
S
be a sequence of r.e. 's defined on a
{X}
(n,
A,
with values on
P)3
is a complete 3 separable metric space.
sponding sequence
{Qn}
(n, A, P)
a sequence
of probability distributions on
{y}
n
of r.e.
'S3
where
Suppose that the corre-
weakly convergent to a probabiUty distribution
on
(S, S)
Q.
(8, S)
is
Then 3 there exists
with values on
(8, S)J
and
such that the following conditions are satisfied:
it)
infinity.
the probability distribution of Yn
For each
n ~ 1
d(Xn ,Y n )
converges to zero in probabi U ty as
n
is
tends to
Q.
58
Proof:
each
n
1,
~
let
n
{F }
m
1,
~
be a Q-continuous Skorokhod partition of
U
P as in Theorem 3.2.1.
and the partition
n
{G }
m
and
{G }
m
are determined by
applied to each
r.e.
Un •
P).
(Q,
n
For each
(Q , P).
{F }
Similarly,
n
m
The Skorokhod construction
determines by the corollary to Theorem 3.2.2, a
with probability distribution
Y
Q.
We now proceed to show that
{Y } satisfies ii).
n
the sequence
m~ 1
Let us recall that in the Skorokhod construction for each
[0,1]
partition of
6.
~1
, •• "
i
m
For
denote the measurable functions, introduced in
Definition 3.1.5, determined by the pair
and
S.
denote a uniformly distributed r.v. associated
n
Xn
with the r.e.
n
P
Let
(Q, P)
was determined by
a
and we denoted by
P.
the elements of the partition associated with
Further-
m
more, recall that by the remark following the proof of Theorem 3.2.2 we
had; for each
m
~
1
and all choices of
i , .•. ,i :
m
1
[F (s},G (8))
0.2.32}
m
m
8
€
S
i
By (3.2.3) and the properties of the sequences
1
i .
, •.. , m
n
{Fm}m->1'
n
{Gm}m->1
we have:
s
(3.2.33)
U (w)
n
n
G (X (w».
m n
If we now observe that the intervals
form also a partition of
assumption that
plies that
X
n
U
n
for
s
€
S
i 1 ,··· ,i m
it follows from (3.2.33) that the
is an interior point of any such an interval im-
should belong to the corresponding element
Hence we can write:
(3.2.34)
[0,1)
[Fn(s},Gn(s»
m
m
59
Hence, from (3.2.32) and (3.2.34)
(3.2.35)
P({w: X (w) r/. Si
n
.} n {w: U (w)
1, .•
,1
P(w: U (w) t [Fn(s),Gn(s»
n
m
m
where
s
is an element of
S.
1
E
n
m
n
6.
1
, .• ,i
})
m
1
{w: U (w) E (F (s),G (s»}
n
m
m
i •
1
, .•• , m
The term on the right on (3.2.35) is the probability that a uniform r.v. belongs to the intersection of an interval with the complement
of another interval.
It can be easily checked that the value of this
probability is given by:
(3.2.36)
(G (s)-F (s»
m
m
- max(O,(min(G (s) ,Gn(s»-max(F (s),Fn(s»).
m
m
m
m
On the other hand by Lemma 3.1.4:
lim
Fn (s)
m
n+cx>
lim Gn(s)
m
= Fm(s)
=
G (s).
m
Therefore, it follows that:
lim P({w: X (w) , S.
n+cx>
n
1
1
, .. ,
i
} n
m
{w: Un (w)
We now observe that given any three events
space we always have:
(3.2.37)
Using (3.2.37) we can write:
(3.2.38)
E
6 i , .. , i })
m
1
A , A , A3
2
1
=
O.
on a probability
60
If we now recall that
U,
n
struction to
Y
was obtained by applying the Skorokhod con-
n
it follows by Lemma 3.2.1 that the second term on the
right of (3.2.38) is zero.
Furthermore, we have just shown that the
first term converges to zero as
(3.2.39)
n
tends to infinity.
Hence, we have:
lim P({w: X (w) i Si
1 } n {w: Y (w) E S
})
n ~ 00
n
1"" m
n
i 1 ,··,im
Let now
E >
0 be given.
Choose
~
m
1
(~)m < E.
such that
= O.
We can
write:
(3.2.40)
> E}) ~
P({w: d(X (w),Y (w»
n
n
> (~)m}).
P({w: d(X ,Y )
n n
On the other hand, we have that:
{w: d(X ,Y ) > (~)m}
n n
(3.2.41)
c
U{w: X i S
1 }n{w: Y ES
i }
n 11 ", m
n 1 1'" m
The last union extends to all sets
S
1 , ... , 1
1
m
in
P.
m
Hence we have that:
P({w: d(X ,Y ) > E})
(3.2.42)
n
I
( 1 1" ••
. )
,1
n
p({w: Yn
E
m
Sl
.} n {w: X i S1
1 ' . . ,1m
n
1 }).
1 ' . . ., m
By (3.2.39) each term of the sum on the right of (3.2.42) converges to
zero as
n
tends to infinity.
E Si
. ), which is independent of n, since
n
1, .•. ,1
m
have the same probability distribution. Furthermore,
dominated by
all
Y 's
n
P(Y
L P(Yn
E Sil, ••. ,i )
m
i t follows that:
(3.2.43)
On the other hand, each term is
lim
n~oo
=1
and hence by the dominated convergence theorem
P({w: d(X (w),Y (w»
n
n
> E})
= o.
o
61
3.3. The atomic and the general cases
As we mentioned before very few changes are required on the proofs
of Sections 2.2 and 2.3 of Chapter II to generalize the results, obtained there, to complete separable metric spaces.
In this section, we
prove a few lemmas that will allow us to make the necessary changes.
We
start by stating a more general version of the definitions of relative
compactness and tightness.
Definition 3.3.1.
Let
probability measures on
{P}
r
of elements of
n
S be any metric space.
(S, S)
r
A family
of
is relatively oompaot if every sequence
contains a subsequence shich is weakly conver-
gent to a probability measure on
(S, S)
(not necessarily an element of
n.
Definition 3.3.2.
Let
S be an arbitrary metric space and
for every
for all
£ > 0
P
in
there exists a compact set
a
is said to be tight if fo
K such that:
P(k) > 1-£
r.
THEOREM 3.3.1. (Prohorov).
spaae.
r
(S, S).
family of probability measures on
r
For a famiZy
r
Let
8
be a oomplete separable metria
of probability measures on
tively aompaat it is neaessary and suffioient that
to be rela-
(8, S)
r be tight.
Theorem 3.3.1 is stated in the way it was proved by Prohorov [1956]
and this will be enough for our purposes.
Varadarajan [1961] extended
the sufficiency of the condition to arbitrary metric spaces.
THEOREM 3.3.2.
Let
atomio probability spaoe
be a sequenoe of r.e. 's defined on the
{X}
n
(Q,
A,
P)
and taking vaZues on
(8,
S).
If
62
the aorresponding sequenae
of probabi Uty measures on
{Q}
n
tight then eveT'!f subsequence of
{X}
n
is
(S, S)
has a further subsequenae which
converges a.e.
all
Proof:
Let
i ~ 1,
in the same way as in the real line case tightness implies
that for each
all
w
Ai
€
i
{A.}
1.
(n, A, P).
denote the atoms,of
there exists a compact set
the sequence
{x (w)}
K
i
Since
such that for almost
k •
is entirely contained in
n
P(Ai»O,
i
But,
each sequence contained in a compact set of a metric space has a convergent subsequence.
Hence, we can apply the diagonal procedure as in
Theorem 2.2.2 to complete the proof of the result.
THEOREM J. J. J.
Let
{X}
n
(n, A, P)
atomic probabitity spaae
S
be a sequence of r.e. 's defined on an
and taking vatues on
is a aomptete separable metria space.
sequence of probabiUty measures
abiUty measure
Q on
(n, A, P)
r.e. 's on
(S, $).
{Qn}
(S,
S) where
Suppose that the corresponding
is weakly convergent to a prob-
Then, if
A
Q
denotes the class of
whose probability distribution on
(S,
S)
is
Q,
we have:
i)
ii)
A
Q
is not empty.
p(Xn,A )
Q
converges to zero as
n
tends to infinity.
The proof is totally analogous to the proof of Theorem 2.2.3.
Corottary.
exists on
In the same conditions, as in the above theorem, there
(n, A, P)
a sequence
{Y}
n
r.e.'s, with probability distribution
verges to zero in probability as
n
of identically distributed
Q,
such that
d(X ,Y )
n n
tends to infinity.
con-
63
Proof:
See proof of corollary to Theorem 2.2.3.
Remark on notation:
on
Sand
p
Recall that we use
d
to denote the metric
to denote the metric on the space of r.e. 's corresponding
to convergence in probability.
We consider now the general case.
space,
{A. }
let
1
Section 2.3
A0
denote the atoms of
= Ui~l
('1, A, P)
Let
A, P)
(0. ,
Since the cases
Ai'
P(A )
O
be a probability
and put as in
=0
P(A )
O
and
=1
were already discussed we shall assume here that:
(3.3.1)
We introduce again the two measures
Pi
and
P
on
2
('1, A)
given by:
o)
P(A )
o
P(BnA
(3.3.2)
P (B)
1
(3.3.3)
P (B)
2
=
=
non-atomic respectively.
(S, S)
If
P
2
('1, A, P )
1
E
A.
and
('1, A, P )
2
X is a r.e. defined on
we will denote by
bility measures induced by
and
B
P(A~)
The probability spaces
values on
all
p(BnA~)
X on
Q,
(S, S)
Q(1)
and
are atomic and
A)
('1,
Q(2)
with
the proba-
in correspondence to
P, Pi
respectively.
Lemma 3.3.1.
Let
{X}
n
be a sequence of r.e. 's on
('1,
A,
P)
and
suppose that the corresponding sequence of probability distributions
on
(S, S)
Proof:
is tight.
Then, both
Totally analogous to the proof of Lemma 2.3.1.
are tight.
64
The result contained in the next lemma can be quite useful when we
deal with weak-convergence from the point of view of continuity sets.
Although we found no reference to it in the literature the result is
probably known and its proof is relatively simple.
Lemma 3.3.2.
Let
S be a separable metric space,
(5, S).
be any two probability measures on
F
1
Let
Q
1
and
and
F
2
Q2
denote
the fields of Ql-continuity sets and Q2-continuity sets respectively.
S
Then,
is the smallest cr-field containing the field
P~oof:
number.
radius
(3.3.4)
Let
x be any element of
We will denote by
r,
B(x,r)
5
and
r
F = F1
n
F2 •
be any positive real
the open ball of center
x
and
that is:
=
B(x,r)
Recall that for
Ac 5
{yES: d(x,y) < r}.
we write
oA
to denote the boundary of
A.
is easy to see that:
(3.3.5)
oB(x,r)
Let now
X
o
c
E 5
{y: d(x,y) = r}.
and
r
O
countable number of values of
0
>
r
be given.
There exists at most a
for which:
(3.3.6)
A similar statement can be made for
Q2'
Hence from (3.3.5) and Definition 1.1.4, it follows that for all
but a countable number of values of
(3.3.7)
r < r O we have:
It
65
Hence, we can choose a sequence
for all
B(xO,rn)
n
~
1,
€
F1
n
limn~
F
2
r
n
= rO
of real numbers,
n
and such that for all n ~ 1,
{r }
,
Therefore, we have that:
=
(3,3.8)
(3.3.8) shows that every open ball of
generated by
F1
n F •
2
Therefore, it follows that this a-field contains
the a-field generated by the open balls.
rable
S,
(n,
A,
P)
{Q(2)}
n
(S, S).
Let
Since the latter, for sepa-
S the proof of the lemma is complete.
coincides with
Lemma 3.3.3.
S belongs to the a-field
{X}
n
be a sequence of r.e. 's defined on
(S, S).
with values on
Let us denote by
{Qn}'
{Q~l)}
the three sequences of probability measures induced by
and
{X}
on
n
Then the weak convergence of any two of the sequences implies
the weak convergence of the third.
Furthermore, if
denote the respective weak limits we have for all
Q,
B
€
Q(l)
and
Q(2)
S:
(3.3.9)
We observe first that if any two of the sequences is tight the same is
true for the third.
This is a consequence of Lemma 3.3.1 if the pair
{Q }.
assumed tight includes
(3.3.10)
for all
Q
(B)
€
S,
n
B
n
Otherwise, recall that:
=
all
n
tight the tightness of
~
1.
{Qn}
are
Hence, if both
follows from the fact that the union of
two compact sets is compact.
Assume now that
Q
n
converges weakly to
Q and
Q(l)
n
converges
66
weakly to
Q(l).
It follows from (3.3.10) that
all sets which are continuity sets of both
is tight every subsequence of
sequence.
and let
{Q~~)}
Let
converges for
Q and
Ql.
contains a weakly convergent sub-
be a weakly convergent subsequence of
QO be its limit.
It follows from (3.3.10) that for all
B
which is both a Q-continuity set and a Ql-continuity set, we have:
=
Therefore all subsequential limits of
{Q(2)}
coincide on a class of
n
sets which is, by the previous lemma, a field which generates
it follows that
{Q(2)}
n
is weakly convergent to a limit
validity of (3.3.9) for all
THEOREM 3.3.4.
(n, A, P)
Let
with values on
metria spaae.
B
{X}
n
Q(2)
A
Q
iJ
(8, S)
where
denotes the alass of r.e. 's on
iiJ
A
Q
8
is a aomplete separable
{Qn}
(n, A,
P)
of probQ.
Then
whose probability
Q we have:
is not empty.
P(Xn,A )
Proof:
and the
be a sequenae of r.e. 's defined on
Suppose that the aorresponding sequenae
distribution is
Hence
S follows.
€
ability measures is weakly aonvergent to a probability measure
if
S.
Q
aonverges to zero as
n
tendS to infinity.
The proof follows the same lines of the proof of Theorem
2.3.1 of Chapter II.
CHAPTER IV
SOME RELATIONSHIPS BETWEEN THE METRICS LAND
p
In this chapter we will look into some relationships between the
Levy-Prohorov metric
L
(Definition 1.1.5) and the metric
p
(see
1.1.4) associated with convergence in probability.
4.1. The lower bound for
p
We will begin by stating a result which is a consequence of the
definitions of
Land
p
and which, we believe, was first proved by
Prohorov [1956].
Lemma 4.1.1.
ability space
Let
X and
(n, A, P),
Y be any two r.e. 's defined on a prob-
with values on
the probability measures induced on
Proof;
S.
Let
£0
>
(S, S)
(S, S).
by
Let
P
x
X and
Y respectively.
be given and let
F be a closed set in
F£Q
to indicate the set:
{XES: d(x,F) s EO}
we can write:
=
Hence, it follows that:
Py be
p(X,Y)
As before we write
P(XEF)
and
P(XEF & Y€F
£0
) + P(X€F & YiF
£0
).
68
But, on the other hand:
{w:
Since
X(w)
€
F}
£0 > p(X,Y)
n
{w: d(X(w),Y(w»
c
it follows from the definition of
P({w: d(X(w) ,Y(w»
p,
>
EO}.
that:
> EO}
Hence, we can write:
P(X€F)
Therefore, it follows from the definition of
L
(Definition 1.1.5),
that:
Since the last result is true for all
the proof of the
o
lemma is complete.
Lemma 4.1.1 shows that given two probability measures Q and Q on
1
2
(S, $), L(Q1,Q2)
is a lower bound for the distance in probability be-
tween r.e.
IS,
and
The following result due to Strassen [1965J shows that this
Q2.
defined in some probability space, whose margina1s are
lower bound is always attained when
5
Q
1
is a complete, separable metric
space.
THEOREM 4.1.1.
spaae and
Q
1
and
(Strassen).
Q
2
Let
S be a aomplete separable metria
be two probability measures on
there exists a probability measure
A
on
S
and Q2 suah that for every pair of r.e.'a
probability distribution is
A,
P(X ,X )
l 2
x
S
(5, $).
with marginals
(Xl' X ),
2
= L(Ql,Q2).
Then,
Q
Whose joint
1
69
For the proof, see 5trassen [1965], Theorem 11, pages 436-438.
We will now look at this result from a different point of view.
Let
(Q,
fined on
A,
P)
be a probability space,
(Q, A, P)
with values on
probability measures on
E be the class of r.e.
(5,
S)
and
de-
G be the class of
which are admissible for
(8, S)
IS
(Q, A, P).
Recall that since we do not distinguish between random elements which
are equal a.e., each element of
r.e. 's.
the class
Let
Furthermore, if
P)
is non-atomic,
of all probability measures on
Z(S)
Q
1
A,
(Q,
E is in fact an equivalence class of
Q2 be two elements of
and
denote respectively the classes of r.e.
tions are
Q
1
and
Q2'
close to the lower bound
ample, that when
S
(5, S).
G and let AQ
and A
Q2
1
whose probability distribu-
X E: A
For a given
tions for the existence of
IS
G coincides with
Q1
we will look for condi-
for which p(X,Y) is arbitrarily
Q
2
L(Ql,Q2)' He now shot... , by means of an ex-
Y E: A
is the real line and
(Q,
A,
P)
is non-atomic, the
construction used in Chapter I I (using the inverse of a d.f.) does not,
in general, produce a r.e. for which the lower bound is attained.
Therefore, if the metric
sequence
{Y }
n
p
is used as a criterion of optimality the
constructed in Theorem 2.1.3 is not necessarily the
"best" one possible.
Example 4.1.1.
Let
n be the closed interval
a-field of Borel subsets of
[0,1]
Let the d.f.
G
2
be given by:
if
x
(4.1.1)
IS
G
1
=
and
{~
if
if
°x
<
S
and
0
x s I
> I
[0,1],
A be the
P be the Lebesgue measure.
70
(4.1.2)
Let
Q1
and
respectively.
€
>
0
{+
=
Q
2
if
x < 0
o:s; x < 1/4
x ~ 1/4.
if
if
be the probability measures determined by
To determine
L(Q1,Q2)
G1
and
we need to find the values of
for which the inequality:
(4.1.3)
F of the real line.
is satisfied for all closed subsets
It is easy to see that for any closed subset
F
of
1
R
we have
that:
= Q2(F
~
n {O,%})
€
Q1«F n {O,%}) ).
Hence it follows that to calculate
L(Q1,Q2)
attention to the three nonempty subsets of
we can restrict our
{O,%}.
If we write for
those sets the inequalities given by (4.1.3) and we use (4.1.1) and
(4.1.2) we will get that
Let now
L(Q1,Q2) =
3
i'
X be the uniformly distributed r.v. given by the iden-
tity map from [0,1]
to the real line.
From the definition of the 1n-
verse of a d.f. (Definition 2.1.4) we have that:
(4.1.4)
Y(x)
=
-1
G (X)
2
o :s; x < 2/3
2/3 < x :s; 1.
=
Hence, we can write:
P({w: Ix(w)-Y(w)I
+ P(Y
=
> €})
=
& X>e)
P(y=O
1/4 & Ix - 1/41
>
E).
G2
71
Therefore, from (4.1.4) and the definition of
= P([O,~]
P({w: Ix«(v)-Y(w)I > e})
+
X it follows that:
n (e:,l])
211
P«J,l] n ([0'4-e:) u (4+e:,1]».
Hence, it follows that:
(4.1. 5)
To obtain
we equate the right hand side of (4.1.5) to
p(X,Y)
solve the resulting equation.
3
8·
=
P({w: Ix(w)-Y(w)I > e:})
The result,
17
36
is clearly larger than
However, it can be easily checked that the lower bound,
attained for the r.v.
=
We will now show that if
L(Q1,Q2).
YE A
Q2
3
is
8'
Z defined by:
Z(w)
there exists
and
e:
3/8 ~ w ~ 17/24
10/4
{
otherwise.
(Q, A, P)
for which
is non-atomic for each
X
E
A
Q1
is arbitrarily close to
p(X,Y)
We are going to need the following known results about non-
atomic spaces.
Lemma 4.1.2.
Let
A be any event on a non-atomic probability
O
A, P).
Let
q1,···,qn
space
I~=l
(n,
qi
~
Then, there exist
P(Ao)·
B ,B ,· .• ,B ,
n
1 2
ppoof:
B.
1.
For
€
A,
n = 1,
Neveu [1965], page 18.)
induction.
P(B )
i
= qi'
be real numbers such that:
n
disjoint subsets of
~
AO'
i=l, ••. ,o.
all
this is a well-known result.
The result for
qi
n
(See, e.g.,
can be easily proved by
0
72
Lemma 4.1.3.
(S, S)
Let
Q
1
and
with finite support.
[SxS, SxS]
1
Let
whose marginals are
defined on the non-atomic space
bution
Ql'
Q
2
be two probability measures on
A be a probability measure on
Q
1
and
(Q,
A,
there exists a r.e.
Q1
and
Q
2
Let
{sl, ... ,sn}
respectively.
P)
Then, for any r.e.
X
with probability distri-
Y with probability distribution
and such that the joint distribution of
ppoof:
Q2'
(X,Y)
{51" ..
and
is
,s~}
Q
2
A.
denote the supports of
Assume that:
Q1({Si n
=
Pi
1 ::;; i ::;; n
Q2 ({sj})
=
qj
1 ::;; j ::;; m
A«si ,sj»
=
Aij
1 ::;; i ::;; n,
Clearly, from the assumptions about
1 ::;; j ::;; m.
Q1' Q
2
and
A it follows
that:
m
I
A
ij
=
Pi
1 ::;; i ::;; n
I
A
ij
=
qj
1 ::;; j ::;; m.
j=l
n
i=l
The construction of the r.e.
Y can now be accomplished by
applying the previous lemma to each one of the sets
X
-1
({s.n
1.
with the
o
THEOREM 4.1.2.
on
(8,
(Q,
A, P)
1
S)
whepe
Let
8
Q
1
and
Q
2
be any two ppobability meaBupes
is a aomplete., sepapable metPia spaae.
be a non-atonna ppobability space and as befope let
Let
A
Q1
By support of a probability measure we understand the smallest
closed subset of S which contains all the mass.
and
73
A
Q2
denote the olasses of roe. 's on
tributions are
and
respeotively.
P(X,A
Q2
Proof:
a sequence
Let
X
A
Q1
€
Xn's.
n
Let
Clearly
Then, for any
=
Since
S
is complete there exists
converges aoe. (and hence in proba-
{x }
n
{Q1n}
{Q1n}
denote the probability distributions of the
converges weakly to
there exists a sequence
port such that
whose probability dis-
P)
of simple r.e. 's (roe. 's taking at most a finite num-
{x }
X.
)
be given.
ber of values) such that
bility) to
(n, A,
Q1'
On the other hand,
of probability measures with finite sup-
{Q2n}
is weakly convergent to
{Q2n}
Q2'
In fact, the set of
probability measures with finite support is dense in the class of all
probability measures in
page 440)
(See, e.g., Parthasarathy [1967],
Recall now that, by Strassen's result, given any two proba-
bility measures
R
1
and
R
2
[SxS, SxS],
A on
measure
(S, S).
on
(S, S),
R and R and such
2
1
whose joint probability distri-
whose marginals are
that for every pair of r.e. 's
(Zl,2 2 )
P(Zl,Z2)
= L(R1 ,R2).
bution is
A,
we have:
denote by
A
n
the probability measure on
this way with the pair
bution
A.
(4.1.6)
But,
n
Q
2n'
X
n
n
{Y }
n
n
and such that
(X ,Y)
n n
n
~
1,
let us
associated in
Q
2n
and
there exists
Yn ,
have a
with prob-
has JOoint distri-
1:
=
is a sequence of r.e. 's on
lity distributions,
Q
1n
~
n
[sxS, SxS] ,
Hence, it follows that, for all
p (X ,Y )
For each
Since both
finite support, by Lemma 4.1.3 given
ability distribution
there exists a probability
{Q2n}
(n, A, P)
are weakly convergent to
whose probabiQ2'
Hence by our
74
main result of Chapter III:
(4.1.7)
lim
n +
~
P(YntA )
Q2
On the other hand.. •
s~nce
Q1
and
Q2
= O.
{Q 1n }
are weakly convergent to
respectivelYt it follows that:
(4.1.8)
=
FinallYt by the triangle inequality we can write:
p(X)A
Q
2
p(X,X) + p(X tY ) + p(Y )A ).
n
n n
n Q2
)
Therefore, by (4.1.6):
p(X)A
Q
)
p(X)X) + L(Q1 ,Q2 ) + p(Y ,A ).
n
n
n
n Q2
2
Hence) by (4.1.7) and (4.1.8) it follows that:
P(XtA )
Q2
Since
X
n
lim
$
n+
converges to
oo
p(X,Xn ) + L(Q1 tQ 2)'
X a.e. it follows that:
Since the reverse inequality is always true by Lemma 4.1.1 t the
o
proof of the theorem is complete.
Remark.
each
X
The theorem above does not permit us to conclude that for
A
there exists Y € A
for which
Q1
Q2
Theorem 4.1.1 says that if we consider in the space
€
E of r.e. 's the
classes of equivalence determined by their probability distributions t
elements of the same class are at the same distance from any other
class) provided that the probability space is non-atomic.
75
The following example shows that this result fails in general.
ExampZe 4.1.2.
1
n = [O,lJ
Let
A
and
1
[0'3]
and all Borel subsets of
sure.
Since the mass of the atomic part is smaller than
easily seen that for any
P will be the Lebesgue mea-
° S a S 1,
a,
an event with probability
(3,1].
be the a-field formed by
a.
there exists in
~
it can be
(n, A, P)
Consider the following probability rnea-
sures on the real line:
=
Q1 ({I})
=
%;
(4.1.9)
Q
1
Both
Q
2
and
=
~
Q2({l})
=
3/4.
(n, A, P)
are admissible for
and by the same
argument used in Example 4.1.1, we can say that to calculate
L(Ql,Q2)
we need only to consider the nonempty subsets of
{O,l}.
However, it is
easy to see that
(4.1.3)
is not sat-
isfied for all
Let now
{I}
E >
is the only set for which
0.
X be given by:
-- {01
X(w)
otherwise.
Observe now that any set in
(n, A, P)
to contain the atom
Hence any r.v.
but ion
3
L(Ql,Q2) = 7; - ~ = ~.
Hence, it follows that:
Q
2
has to be equal to
1
p(X,Y)
whose probability is
1
;;::
Therefore, for any
1/3.
Hence, it follows that:
p(X,A )
Q2
;;::
1/3.
On the other hand, if we consider the r.v.
has
Y with probability distri-
[0'3]'
in
3/4
X'(w)
=1
- X(w),
76
X'
€
A
Q1
and we will show that:
p (X ',A
Q2
In fact, let
) = 1/4.
y
be
defined by:
Y(w)
Clearly, for any
{w: IX'(w)-y(w)1 ~
Remark.
€}
0 <
€,
l
{O
=
€ <
= (~,f].
os
w s 3/4
otherwise.
1,
we can write:
o
The result follows.
Let us consider for
Q
and
1
Q
2
in
G,
the quantity
La
defined by:
if and only if
Q
1
= Q2.
Furthermore, it follows from Theorem 3.3.4 of
Chapter III that a sequence
only if
Lo(Qn,Q)
{Qn}
G converges weakly to Q if and
in
converges to zero as
It is not difficult to show that
n
La
tends to infinity.
satisfies the triangle in-
equality and it follows from Theorem 4.2.2 that for non-atomic probability spaces
La
coincides with
be used to show that
L.
However the previous example may
L
is not necessarily symmetric.
O
We will make
use of Example 4.1.2 to show that the standard procedures to symmetrize
La
(e.g. maximum, arithmetic mean) fail to preserve the equivalence
with
n
~
L.
(Q,
Let
A, P)
be as in Example 4.1.2 and define, for each
1:
I
= -23 +n
Clearly,
Q
n
€
=
G for all nand {Qn}
converges weakly to
by:
Q({O})
=
2/3
Q({l})
=
1/3.
Q,
given
77
Let
Y
€
A be defined by:
Q
if
=
Y(w)
{:
zero as
n
1
w
~
1/3.
0
~
Example 4.1.2 for any
must contain the atom
{X },
n
~
A , X~1({0})
Qn
Hence, it follows that it is impossible
By the same argument used in
to choose a sequence
if
1/3 < w
X
n
tends to infinity.
€
A
Qn
such that
X
n
p(X ,Y)
n
converges to
does not converge
Therefore,
to zero and hence the symmetrized
€
L will not be equivalent to
O
L.
4.2. Weak convergence and equivalent probability measures
In this section, we extend to metric spaces a result of
Padmanabhan [1970] and show how this extension allows us to use
define a metric in the space of r.e.
IS
which is equivalent to
Recall that given two probability measures
surable space
(Q, A)
respect to
(Q«
P(A) = O.
tions
P«
P
Q(A)
=0
Two probability measures
Q and
Lemma 4.2.1.
space
if
(n, A, P).
Q«
Let
for every
P
and
Proof:
equivalent to
in the same mea-
A
€
A,
for which
are satisfied are said to be equivalent.
P
A and
B be any two events on a probability
A necessary and sufficient condition for the
is that
Q on
(n, A)
for which
P(A6B) > O.
Suppose first that for some probability measure
P,
p.
Q for which both condi-
existence of an equivalent probability measure
Q(A) ; Q(B)
to
Q is absolutely oontinuous with
we say that
P)
P, Q,
L
we have:
o<
Q(A); Q(B).
IQ(A)-Q(B)!
Then we can write:
S Q(A6B)
Q,
78
Hence,
Q(~B) >
P(~B)
OJ
0
>
now follows from the definition of
equivalence.
To prove the converse, note that
=
A~ B
P(~B) >
Hence,
Assume that
since
(A-B) u (B-A).
implies that either
0
P(A-B) > O.
P(A-B)
Observe now that if
P is equivalent to itself.
>
or
0
peA)
~
P(B-A)
PCB)
>
O.
we are done
It only remains to consider the case
= PCB).
peA)
Define for each
=
(4.2.1)
Since
P(A-B)
>
C
A
€
=
P(C/A)
P(CnA)
peA)
peA) is strictly positive and hence
0,
Po
is we11-
defined.
Define for each
C
A:
€
Pa(C) + P(C)
(4.2.2)
Q(C)
=
2
Q is a probability measure on
(n, A)
and
Q is equivalent to
P.
On
the other hand:
Q(A)
=
Q(B)
=
(4.2.3)
Since
P(AnB c ) > 0
PO(A) + peA)
2
=
1 + peA)
2
c
[1 + PCB) - P(AnB )]
IR(AnB) + P(B)l 2 =
peA)
[peA)
~
2
I
and
peA)
= PCB)
it follows from (4.2.3) that:
Q(A) > Q(B).
THEOREM 4.2.1.
probability space
Let
(n, A,
X and
P)
a separable metric space.
tinct, that is:
P({w: X(w)
y
be any two r.e. 's defined on a
and taking values on
(S,
S) where
S
Suppose also that the two r.e. 's are dis-
= yew)})
<
1.
Then, there exists a proba-
is
79
bility measure
respeot to
Q on
(St S)..
Let
d
Since
{Ai}
X
that~
with
In
induoe the same probability mea-
Y~
= Y a.e.
£0 > 0
S.
Since
X and
Yare
such that:
> EO}) >
P({w: d(X(w),Y(w»
S
and
X
denote the metric on
distinct there exists
(4.2.4)
suoh
with respeot to all probability measures equivalent
to P it follows that
Proof:
P~
and Y have different probabi Uty distributions.
Q.. X
other words if two r.e. 's..
sures on
Q equivaZent to
(n t A)t
O.
is separable there exists a countable, disjoint collection,
of elements of
St
whose union is
S and such that t for each
it
diam(Ai ) < £0'
Hence, it follows that:
00
I
=
(4.2.5)
i=l
P(X- 1 (A ) n {w: d(X,Y) > £0»
i
From (4.2.4) and (4.2.5) it follows that for some
P(X- 1 (A
i
) n {w: d(X(w) tY(W»
>
EO})
Ai
>
o
we have:
O.
o
On the other hand, since
diam(Ai ) < Eat
o
Therefore, it follows that:
(4.2.6)
P(x- 1 (A
i
) n
(y- 1 (A
o
i
»c)
>
o.
0
By the previous lemma, there exists a probability measure
(n t
A) t
Q equivalent to
Pt
such that:
Q(Y
-1
(Ai» •
o
Q on
80
Or, equivalently, we have:
Q({w: X(w) e A. })
=
~O
o
The result follows.
Definition 4.2.1.
Let
to be a pseudo-metria in
E be any set.
E whenever, for all
1)
d(x,y);;?: 0
and
2)
d(x,y)
=
d(y,x).
3)
d(x,y)
~
d(x,z) + d(z,y).
Remark.
d(x,x)
Every metric on
E
x
{de: e e 0}
= O.
d,
d(x,y) = 0
does not necessarily
of pseudometrics in a set
separating if for every pair
such that:
on a set
(x,y) e ExE:
= y.
A family
de
is said
is a pseudometric but the converse is
not true since for a pseudometric
imply
d: ExE ~ R1
A map
E,
de(x,y) >
O.
(x,y)
E
ExE,
x f y,
with
there exists
Given a separating family of pseudometrics
the topology on
generated by the sets:
E
{{y: de(x,y) < d: x e E, e
is called the topology induced on
Clearly, since the family
E is said to be
E
0
E
>
E by the family
{de: e e 0}
O}
{de: e e 0}.
is assumed to be separating, the
topology induced by this family, is Hausdorf.
Definition 4.2.2.
Let
(n,
A, P) be a probability space and let
E denote the space of r.e. 's defined on
(S, S).
(n, A)
Let
{p e :
A, P) with values on
e e 0} denote the class of probability measures on
which are equivalent to
define:
(4.2.7)
(n,
=
P.
For each
e e 0
and
(X,Y)
€
ExE,
81
Lernrrn 4.2.2.
V·
family
{de: 6
G,
E:
d
a is a pseudometric in
E.
The
is separating and hence induces an Hausdorf
G}
E:
e
E.
topology on
Proof:
on
For each
Since
(S, S),
L
is a metric on the space of probability measures
the verification that each de is a pseudometric on
straightforward.
E is
The other half of the lemma fellows trivially from
Theorem 4.2 . .l.
THEOREM 4.2.2.
with values on
Let
S).
(8,
{X}
n
Then~
and
(n, A, P)
X be r.e.'s defined on
{X}
aonvergea to
n
and only if for every Px-aontinuity set
A c: S
X in probability if
and every
C
€
A we
have:
P(X- 1 (A) n C)
lim
(4.2.8)
n+
Proof:
=
P(X-1 (A) n C).2
n
ClO
1)
{X }
n
Suppose first that
converges to
X in probabi-
Then, it can be shown (see, e.g. , Billingsley [1968] page 26)
lity.
that, for every PX-continuity set
(4.2.9)
1
lim P(X-\A) 6 X- (A»
n
n+ ClO
A:
=
O.
On the other hand, using the properties of the operation
write for any
(4.2.10)
=
C
E:
6,
we can
A:
Ip(X- 1 (A)nC) - P(X- 1 (A)nC)I
~ P{(X- 1 (A)nC) 6 (X- 1 (A)nC)}
n
P(C n (X- 1 (A)6X- 1 (A»)
n
~
n
P(X- 1 (A) 6 X- 1 (A».
n
The desired result follows from (4.2.9) and (4.2.10).
2
The ''leak convergence of
(4.2.8) with
C
= n.
{P~}
to
Px
is equivalent to condition
82
2)
C
€
Suppose now that for every Px-continuity set
A and every
S we have:
(4.2.11)
n +
Let
£ >
P(X- 1 (A) n C)
lim
O.
=
n
P(X- 1 (A) n C).
00
be given.
By the argument used in the proof of the
existence of Skorokhod partitions (see Appendix) there exists a partition of
S by means of a countable collection
of PX-continuity
{B. }
1.
sets, each one of them with diameter smaller than
Hence. we can
e.
write:
(4.2.12)
P({w: d(X (w).X(w»
n
For each
i
L
P(X -1(B.) n X-1( B.c» •
n
1.
i~l
1.
~
~ e})
it follows, from the fact that
1.
is a P -conX
tinuity set and (4.2.11). that:
=
O.
But on the other hand, the sum on the right of (4.2.12) is domin-
~p(X-l(Bi»
ated by
= 1.
n ~ 1.
for all
lim P({w: d(X (w) .X(w»
n+
Since
£
> 0
n
oo
It follows that:
~ £})
=
O.
is arbitrary, this completes the proof.
The next result is the extension to metric spaces of' Theorem 2.1
of ,Padmanabhan [1970].
CoroZZary.
With the notation introduced in Definition 4.2.2, the
metric topology on
v = {de:
quence
e
€
{X}
n
0}
E,
given by
p,
and the topology generated by
are sequentially equivalent.
converges to
In other words, a se-
X in probability if and only if the
o
83
corresponding sequence
e
E:
converges weakly to
for all
0.
P~of:
In one direction the result is trivial since convergence
in probability is preserved when we substitute
P
by an equivalent
probability measure.
To prove the result in the other direction let
tinuity set and let
(4.2.13)
Clearly
=
Q(C)
C
A, P(C) # O.
E:
Define for
A be a
D
E:
P
x
con-
A:
[P(D/C) + P(D)]/2.
Q is equivalent to
P
and
A is a
Q continuity set.
X
It
follows from our assumptions that:
(4.2.l4)
lim
n -+
P(X- 1 {A) n C)
00
=
n
Since (4.2.l4) is trivially true for sets
C,
with
P{C)
= 0,
the
o
result follows from Theorem 4.2.2.
Remark.
If we denote by
topology induced by
V
T
the p-topology on
E
and by
TV
the
it follows from the previous theorem and by
some well-known results in topology (see Wilansky [1970], page 27,
Theorem 3.1.2) that
THEOREM 4.2.3.
that
TV
Let
C
T.
(S'2, A, P)
be a probability space and assume
A is a separable a-field that is
A has a countable sub-class
F such that A is the smallest a-field containing F. Then there
exists a countable subclass
Proof:
G c V such that
T
G = T.
We will show first that with the assumption of separability
Theorem 4.2.2 holds under the weaker assumption:
loss of generality if we assume that
F
C
E:
F.
There is no
is a field since the field
84
generated by a countable class is countable.
=
lim P(A nC)
n
oo
n+
Let now
erates
B
A,
(A)
n
be a sequence
A and assume that for some A E A:
of sets in
(4.2.15)
Let
A
E
and
£ >
P(AnC)
0
there exists
all
be given.
C
C E
F.
Since
F
is a field and
F
gen-
F such that:
E
P(B
~
C)
s
£/3.
By the same argument used in (4.2.10) it follows that for all
D
E
A:
(4.2.16)
IP(DnC) - P(DnB)I
s
£/3.
On the other hand, we can write:
Ip(A nB) - P(AnB)!
Ip(A nB) - P(A nC)/ + Ip(A nC) - P(AnC)I
S
n
n
n
n
+ IP(AnC) - P(AnB)I .
From (4.2.15) and (4.2.16) it follows that:
all
n
larger than some
N
~
Ip(A nB) - P(AnB)I s
n
£
1.
Hence, it follows that:
lim P(A nB)
n+
Let now
C ,
i
{C }
i
with
oo
n
=
P(AnB)
denote the elements of
P(C ) f 0,
i
all
B EA.
F and in correspondence to each
define a probability measure
= {P(B/C.)
+ P(B)}/2
1
Let
B
Qi,
E
A.
G denote the family of pseudometrics determined by
(4.2.7).
alent.
metric:
We have just shown that
Furthermore, since
'G
and
G is countable,
,
given by:
{Q~} as in
are sequentially equiv'G
can be metrized by the
85
Therefore,
and the two metrics
p
and
o
are equivalent.
We now show, by means of an example, that the space
(E,PO)
is not
necessarily complete.
ExampZe 4.2.2.
fined on
Let
(n, A, P)
{X}
n
be a sequence of independent r.v. 's de-
and assume that the corresponding sequence of prob-
ability distributions is weakly convergent to a non-degenerate d.f.
Fn
X "",X
1
n
a-field with respect to which
is the smallest
are measurable.
Put
F = Un->1 F,
and let G be the a-field generated by F. Let B
n
E
and
nO
x
be a continuity point of
Clearly, for all
n
>
F.
F.
Hence,
for some
F
~
L
nO:
~
P({w: X
n
=
x} n B)
P({w: X
n
~
x})P(B).
It follows that:
(4.2.17)
lim P({w: X (w)
n -+
n
00
~
x} n B)
=
F(x)P(B).
Hence, it follows that the sequence of probability distributions
converge weakly to
by (4.2.13) with
{X }
n
F(x)
C
E
F.
with respect to all probability measures given
is fundamental in the metric
X such that
{X}
n
converges to
(4.2.17) and Theorem 4.2.2 that
and hence
F,
It follows that in the case of a countable
PO'
However, if there would exist
X in probability it follows from
X would have to be independent of
F would have to be degenerate which is a contradiction.
F
APPENDIX
We are now going to review the essential facts about metric spaces
that were used in the previous chapters.
With the exception of the re-
sults related to the Skorokhod partition the material is standard and it
is presented here only for the reader's convenience.
A full account of
these and related results can be found in any good introductory textbook on topology or real analysis (see, e.g., Royden [1963] or Wilansky
[1970]).
The basic concepts (e.g.: open and closed sets; interior, closure
and boundary of a set; dense and nowhere dense sets; etc.) as well as
their properties are assumed to be known.
metric spaces by the letter
d.
If
A is a subset of
As before we will denote
S and the metric on
A,
S we write
closure, the interior and the boundary of
AO
and
S will be denoted by
oA
to denote the
A respectively.
A.l. Separable metric spaces
Definition A.l.l.
tion of open subsets of
Definition A.l.2.
to be a base for
there exists
B
S
€
An open
aove~
of a metric space
S whose union in
A collection
x
€
B
c
O.
is a collec-
S.
B of open subsets of
if for every open set
B such that:
S
S
is said
0 in S and every x
€
0
87
Remark.
It is an obvious consequence of the definition that if
is a base for
S
then every open subset of
S
B
can be expressed as a
B.
union of elements of
Definition A.1.3.
A metric space
S
is said to be second Gount-
S
is separabZe if it contains a
able if it has a countable base.
Definition A.1.4.
A metric space
countable dense subset.
In other words,
exists a countable subset
THEOREM A. 1. 1.
D of
S
S
is separable if there
such that
For a metric space
D= S.
the three condi tions be l(J/;)
S
are equivalent.
i)
S
is separable.
ii)
S
is second countable.
iii)
Each open cover of any subset of
S
has a countab le subcover.
For the proof, see Wi1ansky [1970; page 76].
CoroZlary.
Let
S be a separable metric space and
a-field of the Borel sets of
Then,
S
S.
Let
0
be any positive real number.
can be written as a countable, disjoint union of elements of
S each one of them having diameter smaller than
Proof:
{B}
n
S be the
Consider the open cover of
S
o.
given by
Let
denote a countable sub cover which exists by the previous theorem.
Hence, we have:
(A.1.l)
S
=
U Bn
.
n~l
Furthermore, i t is clear that:
(A.1.2)
diam(B ) < 0
n
all
n
~
1.
88
Now since
sequence
{An}
(A.!. 3)
A
(A.I.4)
UAn
S is a a-field it follows that there exists a disjoint
c B
n
S such that:
of elements of
all
n
=
UB n
n
=
~
1
S.
o
The result follows.
THEOREM A.l. 2.
Let
S be a sepal'ab'le metl'ic space.
Then,
S is
the sma'l'lest a-field containing the open balls of S.
Pl'oof:
Since
S
is separable it follows from Theorem A.I.l that
there exists a countable collection of open balls which is a base for
S.
By the remark following Definition A.l.2 we can say that every open
set in
S
is a countable union of open balls of
that the a-field generated by the open balls of
sets of
S.
Hence, this a-field contains
S.
S
Hence, it follows
contains all open
S since S is by definition
the smallest a-field containing the open sets of
S.
The inclusion in
the other direction is trivial since every open ball is an open set.
0
A.2. Completeness and compactness on metric spaces
Definition A.2.1.
A sequence
is said to be a Cauchy sequence if
m and
n
{x}
n
of points of a metric space
d(x ,x )
m n
converges to zero as both
go to infinity.
Remark.
It is easy to see that every convergent sequence in a
metric space is a Cauchy sequence and also that the converse to this
statement is not in general true.
s
89
Definition A.2.2.
A metric space where every Cauchy sequence is
convergent is called a complete metric space.
Remark.
Completeness is not a topological property in the sense
that two metrics
d
and
1
d
2
can generate the same topology on
(that is, they determine the same class of open sets) and yet
S
S
can be
complete with respect to one of them and not with respect to the other.
Definition A.2.3.
every open cover of
A subset
K of a metric space
S
is compact if
K has a finite subcover.
There are several important properties which are equivalent to
compactness in metric spaces.
We will only state the one below which
was used in relation with the definition of tightness.
THEOREM A. 2.1.
and
Let
be a compact subset of a metric space
be a sequence of elements of k.
{x}
n
{Equivalently
point.
k
Then
{x}
n
S
has a limit
admits a convergent subsequence.)
{x}
n
For the proof, see Wilansky [1970; oage 124).
A.3.
The existence of Skorokhod partitions
THEOREM A. 3.1. (Skorokhod).
Let
and P a probability measure on
tinuous Skorokhod partition of
Proof:
for each
k
S
(8, S).
be a separabZe rootria space
Then, there exists a P-con-
S.
In the proof of the Corollary to Theorem A.l.l, we saw that
~
1,
we can express
whose radius are smaller than
there exists a sequence
k
S
as a countable union of open balls
(~)k+2. Therefore, for each k ~ 1
{xi}i~l
of points of
S
and a sequence
90
(k)
{r i
}i~l
of positive real numbers such that:
s =
(A.3.l)
U
all
We claim now that for each
k
~
1
B(x
(k)
i
i
~
r ,
k
is a P-con-
1,
i
,r)
the number of values of
r
in that interval for
is not a P-continuity set is at most countable.
it follows from (A.3.l) and the choice of the
(A.3.2)
1.
In fact, by the same argument used in the proof of Lemma
3.3.2, for each
which
~
we can choose a real number
such that for all
tinuity set.
i
i~l
rk's that for each
Hence
k
~
1
S ::
It is also clear that
We are now going to express the union in (A.3.2)
as a disjoint
union by, defining:
(A.3.3)
i-I
(k)
U B(x
,r ).
k
j
j=l
=
Finally, we define:
k
=
We observe first that
the diameter of
S.
~1
S
i 1 ,···,i k
i
' ••• , k
() •.• () D
i
is a subset of
is smaller than
(~)
B(x
k
.
.
k
(k)
i
k
,r )
k
and hence
Furthermore, since
the class of P-continuity sets is a field and the open balls
are in this class it follows that
S.
~1'
i
..• , k
is a P-continuity set.
The verification of the other properties of the Skorokhod partition
(Definition 3.1.3) is straightforward.
REFERENCES
ALEXANDROV, A. D. (1940-1943).
spaces.
Additive set functions in abstract
Mat. Sb. 8, 307-348; 9, 563-628; 13, 169-238.
BILLINGSLEY, P. (1968).
Convergence of Probability Measures.
John Wiley & Sons, Inc., New York.
DUDLEY, R. M. (1968).
Distances of probability measures and random
Ann. Math. Statist. 39, 1563-1572.
variables.
Mathematical Foundations of the Calculus of
NEVEU, J. (1965).
Probability.
Holden-Day, Inc., San Francisco.
PADMANABHAN, A. R. (1970).
results.
Convergence in probability and allied
Math. Jap. 15, 111-117.
PARTHASARATHY, D. R. (1967).
Probability Measures on Metric Spaces.
Academic Press, New York.
PROHOROV, Y. V. (1956).
Convergence of random processes and limit
theorems in probability theory.
PTKE, R. (1968).
Theor. Prob. Appl. 1, 157-214.
Applications of almost surely convergent constructions
of weakly convergent processes.
Proc. Internat. Symp. Prob.
Inform. Theor., Springer-Verlag, Berlin.
RENYI, A. (1970).
San Francisco.
Foundations of Probability.
Holden-Day, Inc.
92
ROYDEN, H. L. (1968).
Real Analysis.
Second edition, MacMillan,
New York.
SKOROKHOD, A. V. (1956).
Limit theorems for stochastic processes.
Theor. Prob. Appl. 1, 261-290.
STRASSEN, V. (1965).
margina1s.
VARADARAJAN, V.
spaces.
The existence of probability measures with given
Ann. Math. Statist. 36, 423-439.
s.
(1958).
On an existence theorem for probability
Selected Translations in Mathematical Statistics, Vol. 2.
American Mathematical Society, Providence.
VARADARAJ&~,
V. S. (1961).
~~asures
on topological spaces.
Translations of the American Mathematical Society, Series 2, 1965,
Vol. 48,
American Mathematical Society, Providence.
WICHURA, M. J. (1970).
On the construction of almost uniformly conver-
gent random variables with given weakly convergent image laws.
Ann. Math. Statist. 41,284-291.
WILANSKY. A. (1970).
Waltham,
Topology for Analysis.
t~ssachusetts.
Ginn and Company,