Research partially supported by the Office of Naval Research, Contract
N00014-67-A-0002.
THEOREMS OF ERGODIC TYPE FOR STATIONARY SEQUENCES
WITH MISSING OBSERVATIONS
by
R. M. Loynes *
Department of Statistics
University of North Carolina at Chapel Hill
.
Institute of Statistics Mimeo Series No. 643
August 1969
*
This work was done while the author was visiting from the University
of Sheffield.
THEOREMS OF ERGODIC TYPE FOR STATIONARY SEQUENCES
1
WITH MISSING OBSERVATIONS
by
2
R. M. Loynes
University of Sheffield
I IITROlJUCTION
1.
A~ JD SUf~MARY·
stationary sequence, for which
Birkhoff ergodic theorem
surely (a.s. ) to a limit.
1
s p
<
00,
then
L
p
n
-1
EX
=
n
{x :
Let
n
1,2, ••. }
be a strictly
exists: then according to the
n
(Xl + X2 +
...
If, moreover,
+ Xn )
Elxn IP
converges almost
is finite for some
p,
convergence also occurs.
In certain contexts only some of the
X
n
are available (see e.g.
[5]), and it then becomes of importance to know whether the average of
those
X
which are in fact available converges, and if so to what
n
limit; such results also have an intrinsic interest, as Blum and Hanson
[1] pointed out.
this paper.
Results of this and similar kinds form the subject of
In §§2 and3 a.s. and
L
p
properties respectively are
discussed.
Analogous questions may be asked in the continuous-time case, when
a strictly stationary process
{X(t):
-00
<
t
< oo}
is given.
Indeed,
rather more possibilities suggest themselves: for example, one may have
available values at a sequence of times
n- l
EX(t.)
J
{t },
j
is under scrutiny, or alternatively
a 'larger' set
K,
so that the behavior of
X(t)
may be known on
in which case the limiting behavior of
1
Research partially supported by the Office of Naval Research,
Contract N00014-67-A-0002.
2
This work was done while the author was at the Department of Statistics,
University of North Carolina at Chapel Hill.
2
-1 T
T
faX(t)IK(t)dt
I
K
or something equivalent becomes of interest.
is the indicator function of the set
K.)
Certain applications sug-
gest consideration of a slightly different quantity.
Suppose that
is the union of an infinite sequence of (finite) intervals
it might be appropriate to imagine a new process
averaging
X
y
(1)
over these intervals.
= J'
n
then the ratios
(Here
{y }
n
K:
K
then
n
constructed by
If
X(t)I K (t)dt/ r I K (t)dt
n
)
n
n- l En y
become candidates for investigation; corre-
j
sponding definitions and inquiries may be made in the discrete-time case
also.
Results for the continuous-time case occupy §§4 and 5, again for
a.s. and
2.
L
p
properties respectively.
DISCRETE-TIME: A.S. PROPERTIES.
The most direct approach to these
problems is, superficially at least, via the 'Basic Ergodic Theorem'
on p. 415 of Loeve [4], but it is apparently difficult to show that the
hypotheses of this theorem follow from any reasonable conditions.
Here,
in consequence, the method used by BruneI and Keane [2] will be adopted,
and indeed the results obtained are little more than elaborations of
theirs.
It may be useful to emphasize in passing that this slightly
changes the problem: the original task was to discuss
in operator language
n
-1
ET
k.
~f
n -1 En~.,
or
~
for some specific function
f,
whereas
BruneI and Keane's approach relates to the behavior of this ratio for
arbitrary (integrable)
f.
3
The framework is as follows: on some compact metric space
Borel sets
ex, X,
X E
X,
0
is assumed to be given.
{¢
n
x:
{ ~n
~ .
f mappings
= O,l,Z, ... }
n
n
~
l}
is dense in
is equi-continuous.
that there exists exactly one probability measure
The system
h
~
X,
and also
It then follows
(X, X)
on
which
~.
is invariant and ergodic under
If
with
is assumed to be strictly L-stable in the sense that for some
¢)
X the sequence
t h e set
~
a homeomorphism
X,
is a real-valued function defined on
set of discontinuity points of
X,
let
D
h
be the
h.
In addition to this special framework, there is here, as elsewhere,
T
a measure-preserving transformation.
refers to this space.
A,
(~,
supposed to exist a probability space
P),
on which is defined
LP
Notation such as
= LP(~,A,p)
Although this is phrased in a different language
to §l, in which random variables were used, such inconsistency is perhaps not unreasonable.
On the one hand, previous work has been expressed
in operator language, while on the other, motivation and intuitive content (at least for the probabilist) are clearer in random variable
language.
THEOREM 1.
~(Dh)
= 0,
If
h
and
is a bounded real-valued function on
f
E
LI ,
then for any
I
n
converges a.s. as
PROOF.
n-rCO
y
L
k=l
h(¢ky)f(Tkw)
f*
Under the given condition on
h,
hI' h
Z
X
n
to a limit
continuous functions
E
X, such that
E
such that for
Now if
hI
I
L •
for any
x
= h - E/3
I
E
t
>
°
X hI (x)
and
there exist
~
hI! = h
hex)
Z
~
+ E!3,
hZ(x),
it
4
follows that for all
h' (x)
(2)
::s;
x
X
E
h(x) - <:.j3 < h(x) + sj3
h" (x)
::s;
and
f(h" -
(3)
Given
y
E
X,
tinuity of
W of
h') dll < s.
{~n}
then, the equi-continuity of
h' and
h"
and the uniform con-
imply that there exists an open neighborhood
Y such that, for all
n
and
x
E
W,
(4)
From this point on, the only change needed in the proof of Theorem 1 of
[2] is the replacement of
CoROLLARvl.
If
T
Iy "
Iy , Iy"
is ergodic and if
by
T
h', h, h"
and
respectively.
¢ have no eigenvalues
(other than 1) in common, then
(5)
f*
Thus, in this case, if
n
I
0,
h(~ky)f(Tkw)
1
(6)
~
fhdll
In particular if
-+
T
ffdP
a. s.
is weakly mixing, (5) and (6) are valid.
It will be recalled that
T
is weakZy mixing if
n-l
1 L
Ip(T-kE n F) - P(E)P(F)
n k=O
as
n
-+
00
for each pair
E, F
E
A
I -+
0
5
Brunel and Keane term a sequence
(X,
X,
and a point
y
E
E
Y.
some strictly 1-stable system
~
~(dY) =
(Y) > 0,
cessive integers
CoROLLARY
2.
0,
j
~jy
for which
(BruneI and Keane)
If
of integers uniform if for
{k }
n
~,~)
f
there exists
X such that
E
11
k
E
X with
are the suc-
n
{k}
and
Y
is a uniform
n
sequence then
1 n
- L f(T
n 1
f
converges a.s. to a limit
in
k.
J. w)
1
1 •
The proof requires little more than the choice of
Iy
for
h
in
Theorem 1.
Term the set
Y
~(N) =
is a set for which
K(x) ,
X unsaturated if whenever
E
0,
there exist
~J(x)+l x ¢ Y,
such that
Y is unsaturated.
Suppose
x
Y-N,
E
non-negative integers
~-K(x)-l x ~ Y;
and
J(x), K(x)
where
if
~(Y)
N
c
J(x),
<
1,
are chosen as small as possible
and define
lex)
(7)
1 + J(x) + K(x)
=
1
Now suppose that
a ].l-null set.
~nz
E
at
x,
Y or
x
E
Y,
and as
OO
x ~ Nu U
but that
according as
l
x E Y-N,
elsewhere.
Then for any given
yC
when
,pnx
-00
n
E
if
z
Y or
~j (aY) ;
is close enough to
yC•
Thus
is clearly continuous at points in
in the null set just defined it follows that
the latter is
].l(D
l
)
=
l
yC
o.
x
is continuous
which are not
Y
6
If
~f
~
y ~~
~(dY)
Y is an unsaturated set, with
U_00 ~~j(N),
OO
reguZar.
°
=
and
~(Y) >
ca11 the uniform sequence generated by
y
0,
and
Y
and
Then a regular uniform sequence decomposes uniquely into an
A.J.
infinite union of disjoint, finite, maximal subsets
integers, and each
belongs to precisely one
k.
J
posed labelled in the obvious way, so that
A
A.; the
J.
+
of consecutive
A.
J.
are sup-
lies to the right of
i l
A ••
J.
Now define the smoothed
sequenae in such a situation as
f
{gn}'
where
(8)
g (w)
n
I
=
j€A
n
then the following result holds.
THEOREr12.
If
f
€
l
L
and
{gn}
is the smoothed f sequence associated
fld~ <
with a regular uniform sequence for which
00,
1 n
-n I1
converges a.s. to a limit
g
in
gj
Ll ,
and if the conditions of
g = ffdP
Corollary 1 of Theorem 1 are satisfied,
PROOF.
Let
(9)
where
hex)
l
is defined by (7).
Then
1
~
h
~
0,
Theorem 1 may be applied, and it follows that
where
a.s.
N
n
is the maximum element of
and
(N
-1 n
)L g.
1 J
n
A; provided
n
a non-zero limit the proof will be complete.
~(Dh) =
n/N
n
0,
so that
converges,
converges to
Now it follows immediately
7
that
n/N
(10)
n
-+
rhd]1
)
which is non-zero if
fld]l
plies
<
=
fld]1
<
]ley)
It seems plausible that
00.
<
1
im-
but no proof has been found.
00,
An alternative approach is occasionally possible: if
LP-convergence
can be proved, and is rapid enough, then a.s. convergence will follow.
See for example Theorem 6.2 in Chapter X,c' [.3] "
and~
for slightly dif-
ferent conditions,Prob 1ein 13 on p. 265 of Loeve.
3.
DISCRETE-TIME:
LP
PROPERTIES.
Since the sequence of functions in-
vo1ved is uniformly integrable, the next results follow directly from
Theorem 1, together with its Corollary 2, and Theorem 2.
THEOREM
]l(D )
h
3.
If
= 0,
and
converges to
CoROUARY
h
f*
1.
is a bounded real-valued function on
f ~ LP ,
in the
then for any
LP
norm.
f ~ LP
If
(Brune1 and Keane)
1.
n
L f(T
n 1
f
in the
LP
such that
y ~ X
sequence, then
converges to
X,
norm.
k
i w)
and
{k}
n
is a uniform
8
4.
THEOREM
If
f
LP
E
and
{g} in the smoo1hed f-sequence associated
n
with a uniform sequence derived from an unsaturated set for which
<
00
,
then
n
-1
n
L: g.
converges in the
J
norm to
g.
The following results are based on those of Blum and Hanson [1].
THEOREM
S.
Let
be a sequence of real numbers satisfying
n
2
sup
Ih.l· llh.\ = o(n ),
1 ~ i ~ n 1
1 1
n
O(n) , and
llh.1
(i)
(ii)
1
(iii)
n
{h}
n
1
n
-1
I
1
Then if
T
h.
0+
1
l.
is strongly mixing and
f
E
LP
(11)
l # 0
in the LP-norm, and hence if
n
L h.
1
(12)
f(T\J)
1
I1
ffdP.
0+
n
h.
1
It will be recalled that
P[TnA n B]
PROOF.
.
1n
(13)
L2
as
n
A
for every pair
A, B
E
Because of (iii) one may as well suppose
ffdP
= 0,
0+
P[AJP[B]
T is strongly mixing if
0+
00
one has
=
1
n2
\'
L
h.h. V(i,j )
1
J
and then
9
where
V(i,j)
= ff(Tiw)f(Tjw)dP.
parts, that for which
~
!i-j!
On separating the sum in (13) into two
M and that for which
!i-jl > M,
using (i) and (ii) it follows that the result is true in
LP
extended to other
THEOREM
6.
(i)
Let
n
Ih. I
n
=
0
(I
h.)
1
1
and
1
n
Ilh,l = O(I h.).
1 1
1 1
(ii)
Then if
be a sequence of real numbers satisfying
n
1 ~ i ~ n
and it is
by approximation.
{h}
sup
L2 ,
and
T is strongly mixing and
f
E
LP
n
I1
(14)
h. f(Tiw)
1
-+
n
I
1
JfdP
h.
1
The proof uses the same technique as before.
The only point which
is not completely trivial is in showing that
(~nl h,)-2
~I'1-J'I <M
h.h, V(i,j)
1
1 J
by a constant multiple of
tends to
O.
This, however, is bounded
(2M + 1) • sUPl <_ l. _< n h.·
Enlh,l
• (E h,)-2,
1
1
1
and the use of (i) and (ii) completes this part of the proof.
U:>ROLLARY
I
(Blum and Hanson)
If
T
is strongly mixing and
any strictly increasing sequence of integers, and if
n
-1
n
k·
~l f(T 1 W)
If
converges to
ffdP
in the
f
E
LP ,
{k }
n
is
then
p
L -norm.
A is an infinite subset of the positive integers, term it
deaomposable if it is the union of disjoint, finite, maximal subsets
A.1
10
of consecutive integers; in §2, an unsaturated
compos able sequence.
gn
Write
set gave rise to a de-
n(A.)
= n.,
1
{gn}
is the smoothed f-sequence associated
1
and again, for
define
by (8).
THEOREM 7.
If
f
E
LP
and
with a decomposable subset of the integers, and if
T
is strongly
mixing, then
I
1 n g.
n 1 J
ffdP
-r
in the LP-norm.
PROOF.
Define
h
k
0
-1
if
J
= n.
and replace
while
Nn
Ll hi
n
k ~ A,
if
k
in Theorem 6 by
= n,
E
A. ,
J
N •
n
Then
1 ;::: h. ;::: 0
1
for all
i,
and conditions (i) and (ii) are trivially satisfied.
It will be observed that, although Theorem 7 is stated in this form
so as to show the parallel with Theorem 2, there is in the present case
no need for the subsets
A.
lies to the right of
4.
1
to be maximal: any decomposition for which
A.1
for all
CONTINUOUS-TIME: A.S. PROPERTIES.
i
will suffice.
As far as one of the two formu-
lations possible in continuous time is concerned, that in which a sequence
{t.}
J
of times is involved, no progress of any significance has
been made; results are easily obtained, of course, if the gaps between
11
successive members of the sequence
i f the
form a periodic sequence, or
{t.}
J
are all integral multiples of some unit of time, or finally
t.
J
if LP-convergence occurs sufficiently rapidly, as was observed for the
discrete-time case at the end of §2.
Results for the other kind of situation are straightforwardly
found, only obvious changes being necessary in the arguments of §2.
First, a measurable semi-group of measure-preserving transformations
{T :
t
t
>
O}
{ql'
t'
Next, it is supposed that there exists a semi-group
x,
homeomorphisms on
x,
{ql t x·.
with dense orbit
t
>
O}
A,
p).
t > O}
of
for at least one
and this semi-group is assumed to be equi-continuous and measurable;
the latter condition of measurability is satisfied if
on both sides at
THEOREM
8.
If h
~(Dh) = 0,
=0
t
as the inverse of
and
for every
then for any
ft
0
t
CoROLLARY
1.
n
+
=
is defined for negative
Jhdll
s
Thus in this case if
fhdll
~
0,
and that
{qlt}
have no
s
f* ELI.
is ergodic and if
• JfdP
X,
y E X
eigenvalues (other than 1) in common, then
f*
is continuous
h(ql y)f(T w)ds
to a limit
00
{T }
t
If
(qlt
is a bounded real-valued function on
fELl,
converges a.s. as
x.
qltX
ql-t.)
-1
(15)
(~,
is supposed given on the probability space
a.s.
{T }
t
and
t
12
t
Jo h(!b s y)f(T s w)ds
(16)
In particular, if
{T }
t
~
a.s.
JfdP
is weakly mixing, (15) and (16) are valid.
The definition of weakly mixing in the present context is obtained
from the discrete-time version by making the obvious change of replacing
summation over
k=O
Y E X,
the set of those
CoROLLARY
2.
If
n
by integration over
s=O
to
t.
K of the non-negative real axis be called uniform
Let a subset
if for some
to
s
with
~(Y) >
for which
f EL
I
converges a.s. to a limit
and
f
0 =
~(aY),
and some
y E X,
K is
!bsY E Y.
K is a uniform set, then
in
l
L •
It does not seem possible to find reasonable general conditions
which would imply an analogue of Theorem 2, and it appears easier to
investigate particular cases when the need arises.
13
5.
CoNTINUOUS-TIfViE:
L
P
PROPERTIES
I
For this mode of convergence, it
is very simple to obtain results for a sequence
THEOREM
9.
as
I+
\i-j
Let
{t.}
be a sequence of real numbers such that
J
and let
00,
{t. }.
J
{h}
n
t.-t.+
].
J
be a sequence of real numbers satisfying
(i)
n
Ilh.l
1 ].
(ii)
O(n),
and
(iii)
n
Then if
-1 ~
L h. +
1 ].
{T }
t
is strongly mixing and
-n1 nI1].
h.
(17)
.e..
f(T
ti
w)
+
in the LP-norm, and hence if
f
E
LP
l JfdP
l # 0
n
I
h. f(T
1].
(18)
ti
w)
n
I1
THEOREM
t.-t. +
]. J
10.
00
Let
as
h.
].
{t.}
J
li-j I +
be a sequence of real numbers such that
00,
satisfying
(i)
sup
1 :::; i :::; n
and
Ih.1
.
].
and let
{h }
n
be a sequence of real numbers
00
14
n
n
Ilh.1
1
(ii)
O(I h.).
=
1
1
Then if
{T }
t
1
is strongly mixing and
f E LP
n
I1
(19)
h. f(Tt.w)
1
1
-+
n
IfdP
I1 h.
1.
in the LP-norm.
THEOREM
11.
If
{T }
t
real numbers such that
n -1
"n
f( T w)
6
1
t
{t }
j
is strongly mixing and
t.-t.
J
1
converges to
-+
00
as
ffdP
!i-jl
-+
is a sequence of
f E LP ,
and if
00,
then
in the LP norm.
i
For an analogue to Theorem 7, it is necessary to take up the last
Define a deaomposition of the set
remark of Section 3.
Z in the form
integers as an expression of
finite, and
Ai +
l
lies to the right of
Ai'
define the smoothed f-sequence associated with
{g},
n
UA ,
i
Z
of positive
where each
A.
1
is
Then corresponding to (8),
{t. }
J
and
{A.}
1
as
where
(20)
I
=
f(T
jEA
n
THEOREM
12.
If
f E LP
t
I
w)/
j
and
1.
jEA
n
{g}
n
is the smoothed f-sequence associated
with a decomposition of the integers and a sequence
bers such that
t.-t.
1
J
-+
00
as
li-jl
-+
00,
and if
of real numis strongly
15
i
1 n
n
gj
-+-
ffdP
in the LP norm.
The proofs of these results are exactly similar to those of the corresponding ones in Section 3.
Next, results for the case in which a set
appears.
K of positive measure
The following are immediate consequences of Theorem 8 and its
Corollary 2.
THEOREM
]J
(D )
h
13.
= 0,
If
h
and
f
is a bounded real-valued function on
LP ,
E
then for any
1
~
It
°
h(~
converges to
f*
in the LP norm.
I.
If
f
CoROLLARY
converges to
f
E
LP
andK
s
y
E
X,
such that
X
y)f(T w)ds
s
is a uniform set then
in the LP norm.
Finally, results which parallel Theorems 5, 6, 7, 9, 10, 11 and 12.
16
THEOREM
14.
Let
(i)
a
(ii)
f:
(iii)
t-
sup
be a measurabLe real-valued function satisfying
Ih(s) Ids
f:
l
{T }
t
=
h(s)ds
is strongly mixing and
£
+
Let
(i)
h
=f
f
E
LP
a
ffdP.
be a measurable real-valued function satisfying
sup
Ih(s)
a :s; s :s; t
I
=
O(f:h(S)dS)
and
f:
(ii)
Then if
(22)
2
(t ),
£.
+
t h(s)f(T w)ds
fa
s
THEOREM lS,
0
O(t),
in the LP norm, and hence i f
(22)
=
t.!h(S) I • fat Ih(s)1 ds
:s; s :s;
and
Then if
h
{T }
t
Ih(s) Ids
=
is strongly mixing and
t h(s)f(T w)ds
s
fa
in the LP norm.
O(f:h(S)dS).
f:
h(s)ds
f
E
LP
17
CoROLLARY
measure
I.
If
{T }
t
is strongly mixing, and
then for any
00,
f:
(23)
f
K is a set with Lebesgue
LP
E
IK(s)f(Tsw)ds
f:
IK(s)ds
in the LP norm.
If
{A}
is a sequence of measurable sub-sets of the real line,
n
u.n },
having positive Lebesgue measures
to the right of
Ai'
term
(24)
I
=
n
{A}
is
by analogy with (8), viz.,
n
A
A+
i l
i
admissible and define the smoothed
{A}
f-sequence corresponding to
and if for each
(s)f(Tsw)ds.
n
THEOREH 16.
If
{g}
n
is the smoothed f-sequence corresponding to an
admissible sequence of subsets
nl
n
-+
00
'
and i f
LP
is strongly mixing, then in
n
for which
{A },
n
t
-1 n
gj
-+
J
fdP.
The proofs of Theorems 14 and 15 are straightforward and Theorem 16
follows from Theorem 15 on writing
00
h(s)
=
\'
L
1
l.-1 I
1
Ai
(s).
18
REFERENCES.
[1]
Blum, J.R. and Hanson, D.L. On the mean ergodic theorem for subsequences. Bull. Amer. Math. Soc., ~, (1960), 308-311.
[2]
BruneI, A. and Keane, M. Ergodic theorems for operator sequences.
Z. Wahrscheinlichkeitstheorie verw. Geb., 12, (1969),
231-240.
[3]
Doob, J.L.
[4]
Loeve, M.
[5]
Loynes, R.M. Aliassing in time-series. To appear in Proceedings
of XlI-th Biennial Seminar of the Canadian Mathematical
Congress.
Stochastic Processes.
Probability Theory.
New York: Wiley, 1953.
Princeton: Van Nostrand, 1955.
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Techni ca1 Report
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AUTHOR(S) (Laet nll/lle. firet neme, InltlllO
Loynes, Robert
r~.
, •. 'l'OTAL NO. OF PAGES
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ell.
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13. ABSTRACT
Accordi ng to the Birkhoff and von Neumann ergodic theorems, the arithmetic mean of
tne fi rst
n
tends to 00,
variables in a strictly stationary stochastic process converges as
almost surely and in
2
L •
If not all variables are available, it is
natural to consider the average of the first
related questions.
n
available variables, and closely
Such topics, including the analogous situations in continuous
time, form the subject of the present paper.
DD
FORM
, JAN 64
1473
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n
lJNCLASSIFIEU
,
Security Classification
:"'1-4.--":::';;;;'=':~";;';;;';;~;';';'';';';~----------------'''''-~L~I~N~K:-A~-"T"-~L~IN~K~B:---"'--L:-I~N~K:-C~-"
KEY WORDS
ROL.1t
WT
ROL.E
WT
WT
ROL.E
'---',
limit theorems
ergodic theorems
:
ergodic theorems without semi groups
missing observations.
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