*This research was partially supported by the National Science Foundation.
ASYMPTOTIC FLUCTUATION BEHAVIOR OF SUMS OF
WEAKLY DEPENDENT RANDOM VARIABLES*
by
Wal ter Philipp
University of Illinois at Champaign-Urbana
and
William F. Stout
Universi ty of Il Zinois at Champaign-Urbana
and
Department of Statistics
University of North Carolina at Chapel Hill
Institute of Statistics Himeo Series ilo. 950
September 1974
ASY~PTOTIC FLUCTUATION BEHAVIOR OF sur~s OF
WEAKLY DEPENDENT RANDOM VARIABLES
by
l'lal ter Philipp
University or Tl"li1io1:S at' Champaign "- 'Urbana
and
'fli 11 iam F. Stout
University of Illinois at.'ChiurTpaign - Urbana
and
University of North Carolina at ChapeZ HiU
1.
INTRODUCTION
The purpose of our research is to investigate the asymptotic fluctuation
behavior of sums of weakly dependent random variables, such as lacunary
trigonometric, mixing, and f;aussian.
He present here a brief exposition of
the results obtained and a detailed sketch of the method leading to these
resul ts.
A complete presentation is given in Philipp and Stout (lC'74).
In essence, {xn} are weakly dependent if
as
k
+ m
for each
n
~
I.
By asymptotic fluctuation behavior, we mean
results such as the law of the iterated
of the iterated
logarit~~,
logarit~~,
Strassenis functional law
the upper and lower class refinement of t!'.e
law of the iterated loga:..ith1'l, and Chung;s upper and lower class result for
the maxima of partial sums.
-2We obtain these results by first establishing almost sure invariance
principles.
TIle idea of an almost sure invariance principle is due to
Strassen (1964, 1965).
Strassen pYoves, among other things, that a martingale
with finite variances is with probahility one close to 3rownian
in a sense nade precise in Section 3.
[0,00)
behavior of
Bro~~ian
kno~m,
logarithm [Strassen,
on
The aSYMptotic fluctuation
motion has, of course, been thoroughly investigated.
particular, the upper and lower class refinement of the
logarithm is
~otion
la~l
In
of the iterated
as well as the functional form of the law of the iterated
(l~64)]
and the upper and lower class refinement for the
maximum of Brownian motion up to
ti~e
t [Jain and Taylor, (1973)].
Thus, if
the approximation of the martingale to Rrownian motion given by the almost
sure invariance principle is sufficiently close, then all the above fluctuation
results for Brownian motion also hold for the
nartin~ale.
If we are able to approximate sufficiently closely sums of weakly dependent random variables by a martingale,
above
1 .
ren~arks,
,~e
can then conclude, in view of tte
that these sums are close to
Brol~ian
Consequently, the fluctuation results for
notion with probability
Brol~ian
motion will continue
to hold for the weakly dependent random variables under consideration.
Let us be more specific about this last point.
Let
of random variables, centered at expectations with finite
Put
(1.1)
Suppose
St
= Set)
--
t
x
Ln~t n
{x}
n
be a sequence
(2 + 0)
moments.
-31
lim var[n -2 Sn) =
2
(J
> 0
exists and is positive so that without loss of generality we can and do assume
2
(J
= 1.
Our goal is to prove for a large class of weakly dependent random
variables the basic almost sure invariance principle
1
Set) - X(t) « t
(1.2)
for some
Here
Tl >
X(t)
sequence
o.
a.s.
(We use the Vinogradov symbol «
is standard Brownian motion on
{x
n
} considered.
{~(t)}
suppose we have an arbitrary sequence
{S (t)}
instead of big
and
[0,00)
Tl >
0
0 .)
depends on the
Such a result immediately translates almost sure
fluctuation results from {X(t)} to
function"
Z-Tl
satisfies (1.2).
and hence to
{x }
n
{Sn}.
Indeed,
such that its IIbookkeepinp,
Then the follm\ring four theor.ems are
straightforward consequences of known results for
3ro~mian
motion.
00
THEOREM A: Let {xn}n=l be any sequence of ranOOl'l variahles satisfying (1.2).
Let
$(t)
be a positive, nondecreasing real-valued function.
Then
1
P{S
>
n
2
n ~(n)
i.o.}
=0
or
1
according as
exp (_ !. $2 (t))dt
t
2
f1 till..
OO
converges or diverges.
This result follows from Kolmogorovis test for Brownian motion.
details of the proof see for example Jain, Jogdeo, and Stout (1974).
For the
Using a recent result on the maximum of
Bro~mian
motion due to Jain and
Taylor (1973) He get (again, for the details see Jain. Jogdeo, and Stout (1974)).
THEOREM B: Let {xn } and
~(t)
Mn
be as in Theorem A.
= max l :;';l_n
"<
Put
15.1
•
1
Then
1.
P{M <
n
n2 $-l(n) i.o.} = 0 or 1
according as
J
~l ~2(u)
~ u
exp(-8w- 2 $2 (u))du
converges or diverges.
Next. let
h(t).
K c
J:
t
C[O.I]
€
C[O.l]
be the space of all real-valued continuous functions
I Ihll = sUPO:;;t:;;l
(0.1] , with the supremum norm
Ih(t)1 .
p-e the set of absolutely continuous functions with h(0)
(h'(tl)2dt < 1.
Let
= 0 ,
Let
(1.3)
Sn (t)
= S (nt)
=
Lk<_n t
Define another type of bookkeeping functions
xlc •
fn(t) = f n (t.w)
by
1
(1.4 )
f (t) = (2n log log n) -2 S(nt)
n
~
THEOREM C: Let {xn }n=l be a sequence of random variables satisfying
(1.2).
Then with probability 1 the sequence of functions {fn(t)} defined by (lA) is
-5-
relatively compact in the topology of uniform convergence and has
7.'
as its
derived set.
Indeed, by (1.2) and (1.3),
1
1
2-n
S (t) - X(nt) «(nt)
n
uniform! y in
o~ t
~
:''rite
1.
yn
c
a.s.
n
r;; (t) = X(nt) .
n
Then, of course,
1
II Sn
(1.5)
-
l;n!
I«
"2 n
n
n
> 0 .
11«
n-n .
a. s. ,
Consequently,
1
(n log log n) -211 S n
l;
n
Tneorem C follows now from Strassen!s (1964)
~.eorem
1
whic~
states that the
conclusion of Theorem C holds for the sequence
{ (2n log log n)
--21
r. n (t) ,
n
3}
~
Similarly, (1.2) implies distribution type invariance principles.
THEOREt;1 D:
Let {xn }:=l
be a sequence of random variables satisfying (1. 2).
Then
1
n -I S (t) -+- W(t )
n
in distribution where
Wet)
is standard Brownian
is defined by (1.4).
Indeed, (1.5) inplies that
1
n-I(s n
1; ) -+-
n
0
~tion
on
(0,1]
and
~
n
(t)
-61
in probability.
for
n
~
1
Eut
n
-2
r;
has the same distribution as
n
{N(t) ,Os t s I}
and the result follows (see e.g. Billingsley (1968)
Both Theorems C and
~
~.
25).
have a large number of corollaries spelled out in
detail in Strassenvs paper (1964) and Billingsley's book (lQ68).
In addition to the stationary case, several applications are
~ade
to the
non stationary case where
lim n
fails.
-1
Then our goal is still to prove the basic relation (1.2),
\~lhe!'e
definition of {Set)} is nodified appropriately:
n
S (t)
=L
2
t <
for
sn+l
k=l
Theorems A-D continue to hold, provided the statements are also modified
appropriately.
In Theorem A replace
1
pIsn
n2
>
~(n)
=0
i.o.}
or
1
by
PIS
>
n
s
~(s2)
i.o.}
n
n
=0
or
1 .
Similarly, in Theorem P" replace
1
P{M <
n
by
'2
n
~
-1
(n) i.o.}
=0
or
1
the
-7-
In Theorems C and D redefine
S (t)
n
?
= S(s~n
t) , 0
~
t
~
1 .
In Theorem C, replace (1.3) by
1
=
f (t)
n
(2 5 2 log log s2) -2 S (t)
n
n
n
TIle modifications in the proof of Theorem C are obvious.
1
n -2
In Theorem D replace
~n(t) by s-l S (t)
n
n
For all cases of weakly dependent random variables that we consider the
conclusions of
TIleorerr~
A, B and C are new.
The conclusion of Theorem D,
however, is new only in certain cases.
It will be clear that the present method is rather general and is thus
applicable to many situations other than those considered in this monograph.
We will give an outline of our method in Section 3.
2. STATEMENT OF RESULTS
Let
{n , k
k
~
l}
be a lacunary sequence of real numbers, i.e.
{ a.}
K
Suppose that
such that
AN
~ ~
Put
be another sequence of real numbers.
and that there exists a constant
0 with 0
<
0
~
1
-8-
1-0
(2.1)
~«Ak
.
We consider trigonometric series
= O.
Bk
where for reasons of convenience we put
l'f
A2
f\N
For
t
~
0 put
$
TI1en we have the following theorem.
THEOREM 1: 11ithout changing the distribution of the process {Set), t
we can redefine the process
{Set), t
together with 3rownian motion
{X(t), t
Set) - X(t) « t
for each
~
!...co
2
O}
on a richer probability space
O}
~
~
O}
such that
a.s.
c < 1/16.
Condition (2.1) can quite likely be replaced by a weaker one, say
~ « J\/log
'1<
It is also quite likely that the gap condition could be
relaxed to
combine these
Perhaps, one could also
tw~.:·possibl~·generalizations.
Theorem 1 can be considered as a refinement of M. Weissis (1959) law of
the iterated logarithm under the restriction (2.1).
Also 3illingsley's
functional central limit theorem for lacunary trigonometric series should be
mentioned in this context.
(Billingsley (1967)).
-9-
Finally, it should be remarked that Theorem 1 includes the unweighted
case a k
= 1. In this special case, a much simpler proof can be given.
Recently and independently froM the authors Berkes (1974b) obtained a sir.ilar
result in the unweighted case, making more stringent assumptions on the
{~}
sequence
.
,~
generated by xa ' xa + 1 '
Ynen the sequence
{~(n)}
there exists a sequence
each
t
~
1, n
~
0, A
t
Fl' B
t
denote the a-field
F""
the a-field
m
is said to be
~(n) ~
of real numbers with
0
~-mixing
if
such that for
F""
t+n we have
t
Ip(AB) - P(A)P(B)I
(2.2)
F
a
and
, (1 S a s b < (0)
generated by xm' xm+l '
b
{xn , n ~ I} ,let
Given a stochastic sequence
s ~(n)
peA) .
We assume that
1
""
rn=l
(2.3)
~
"2
(n) < "" •
This is easily seen to imply the existence of the
0
2
li~it
= 1imn>m n- 1 E[rk<n 'k]2
Ne assume throughout that
a
2
> 0 •
the case
a
Hence we assume wi t110ut loss of generality that
a
(2.4)
2
=1
.
Let
Set)
=
rkst
xk ' t ~ 0 .
2
=0
being a degenerate one.
-10-
Then we obtain the following almost sure invariance principle.
THEOREM 2: Let {xn ' n
~
1} be a stationary sequence, centered at expectations
and satisfying (2.2), (2.3), and (2.4).
2+0
E1xl 1
<
Suppose that for some
00 .
Then, without changing the distribution of
the process
standard
{Set), t
Bro~mian
~
{Set), t
~
O} , we can redefine
on a richer probability space together with
O}
motion
0 > 0
{X(t), t
Set) .. X(t) « t
~
~
O}
such that
0/ (12+60)+y
a.s.
for each y > 0 .
REMARK:
If
IIx11100 <
00
then the exponent reduces to
~
+ Y •
Theorem 2 contains one of Reznik's (1968) theorems as a special case.
Independently from the authors, Berkes (1974a) establishes a similar result
under stronger hypotheses.
It might be interesting to remark that if, instead of
2+0
EIxl 1
<
00 ,
only the finiteness of the second moments is assumed then the conclusion
of Theorem 2 has to be weakened to
I
Set) - X(t)
= o(t
log log t)2)
a.s.
This result is due to Heyde and Scott (1973).
At the relatively minor cost of strengthening the assumption (2.3) and a
more complicated proof, Theorem 2 can be considerably generalized.
done in the next four theorems.
This is
Before stating these four theorems, we nake
-11some preparatory
Let
{~n}
remar!~s.
be a sequence of random variables.
Let
mapping fron the space of doubly infinite sequences
real numbers into the reals.
be a measurable
f
( ... ,a_l,aO,a , ..• )
l
of
Put
nn = f (... , ~ n- I' E;;n ,~ n +1' ... ) , n ;:: 1 .
(2.5)
Instead of
~-mixing
retarded strong mixing condition.
generated by the
~
n
~'s
we shall assume that the
satisfy what we call a
Denote, as usual, by
(a :::; n :::; b) •
's
n
We shall
assu~e
b
F
a
that
the a-field
Ip(AB) - P(A)P(B)I : :; a(n t- K)
(2.6)
and all
for all
00
BE F
t +n
.
Here
is a nonnegative constant,
K
depending only on the exponent of the Doments of the
converges monotonically to zero at a certain rate.
n 's
n
The case
anc.
K
a(s)
= 0
gives
t~le
so-called strong mixing condition introduced by r1. Rosenblatt (1956).
Conditions of the form (2.6) occur in the metric theory of Diophantine
approximation (see for example Szfisz (1963) or Philipp (1971), p. 48).
Finally, we shall make no stationarity assumptions.
assume that
nn
can be closely
approxiL~ted
As is customary, we shall-
by
With this notation we 11ave the following theorem:
THEOREM 3: Let
defined by (2.5).
{~n}
be a sequence of random variables and let
He shall assume of the function
f
nn be
and the sequence
{E;;n}
-12-
that
En
n
=0
•
Suppose that there exists constants
0
0
<
S
2 and C
>
0 such that
and
II nn -
(2.7)
for all
as
N+
n,t
~.
= 1,2,3, •.•
n
tn
II 2+6-<
C. t-(2+7/o)
Moreover, suppose that
Finally, assume that
{sn}
satisfies a retarded strong wlxing
condition of the forTI (2.6) with
K
= 0/(11 + 40)
and
a ( s ) «s
(2.3)
-163 (1+2/0)
Define a continuous parameter process
S (t)
=I
nst
.
{Set), t
~
O}
by setting
n
n
Then, without changing the distribution of . {Set), t ~ o}
process
{Set), t
~
o}
we can redefine the
on a richer probability space together with standard
-13-
Brownian motion
(X(t), t
~
O}
such that
1
Set) - X(t)
for e.ach c
<
« t 2-
CO
a.s.
as
t
a~d
(2.8),Theorem 3 contains in
~ ~
1/583 .
Except for
t~e
rates of decay in (2.7)
the stationary case almost all previous results in this direction.
For
example, it contains Reznikis (1968) and Iosifescu's (1968) theorems on the lal:
of the iterated
lo~arith~
for stationary
the convergence rates in (2.7) and (2.8).
~-mixing
random variables except for
The same comment holds for ?eznik's
(1968) law of the iterated logarithm for strong mixing sequences.
in this latter case we only assume a uniform bound on the
n . Reznik either assumes uniform bounds on the
n
(4 + 0) moments.
n 's
n
(2
+ 0)
notice that
moments of
theDse1ves or on the
Theorem 3 also contains recent upper and lower class results of Jain,
Jogdeo and Stout (1974) on functionals of a turkov process satisfying
Doeblinis condition.
Independently of the authors, Berkes (l974a) establishes
a result which is essentially the same as Theorem 3 for the special case that
K
=0
.
Again, except for the rates of convergence in (2.7) and (2.3), Theorem 3
also implies a result of Davydov (1970).
[loreover, it even implies a non-
trivial, though rather weak, remainder term in the central limit theorel'1 for
strongly mixing stationary sequences.
Although strict stationarity was not assumed in Theorem 3, restrictions
consistent with stationarity on the growth of the variance of the partial
-14-
sums of the random variables were imposed.
the
(2
+
0)
In the same vein we ussumed that
moments of the random variables were uniformly bounded.
next two theorems we relax these restrictions.
In the
Eowever, for the sake of
simplicity we shall restrict ourselves to sequences of random variables
satisfying a strong mixing condition without introducing the retardation of
Theorem 3.
For the next two theorems let
{xn } be a sequence of random variables
centered at expectations and with finite
o<
(2.9)
(2 + 0)
moments where
s 2 .
~
Suppose that
(2.10)
as
N+
~
and that
(2.11)
for some
0 S psI .
Ibreover, suppose that
{x }
n
satisfies a strong
mixing condition with
a(k) «k- 300 (1+2/0) .
(2.12)
That is
Ip(AS) - P(A)P(B)I s a(n)
for all
and all
B
co
€
F
t +n
field generated by {xn,a s n s b} .
.
Here, as usual
Fba , denotes the
0-
-15-
Dith this notation we have the
fo1lrn~ing
two theorems.
THEOREM 4: Let {xn } be a sequence of random variables satisfying (2.9).
(2.10). and (2.12).
Suppose that there is a constant
...,L IX 12+ 0
n
Let
~
C
n
C such that
= 1.2;
F be an arbitrary monotonically increasing function.
Suppose that
(2.13)
whenever
(2.14)
for some A •
not exceed
In other words suppose that
~~e left-~and
side of (2.13) does
F evaluated at the left-hand side of (2.14).
Define
Set)
=L
x
k~N
n
for
s~ ~ t
<
Then. without changing the distribution of
redefine
{Set). t
Brownian motion
~
O}
{X(t). t
c < 1/588.
•
{Set). t
~
O} ·we can
on a richer probability space together with standard
~
O}
Set) - X(t)
for each
S~+l
such that
1
Co
r
«t
a.s.
-16-
THEOREM 5: Let {xn } be a sequence of random variables satisfying
(2.12) with p ~ ~. Let a be a constant with
pO' ~
(2.9) -
1/10 •
Suppose that uniformly in M = 1,2, ... ,
(2.15)
as
N -+
Then the conclusion of Theorem 4 renains valid for each
00
c < 1/1232 .
REMARKS:
Conditions (2.13), (2.14), and (2.15) are of a similar structure as
the Ljapounov condition for the central limit theorem for sequences of independent random variables.
Except for the rate of decay of
Philipp (1969) as a special case.
a(k) , Tr..eoren S contains a result of
This result of Philipp's is, to the best of
our knowledge, so far the only law of the iterated logarithm for
sequences of random variables satisfying a mixing condition where monents are
not assumed to be uniformly bounded.
We can considerably relax the retarded
mixin~
.
~
in (2:5). In other words let
nn = "n
case where
condition in the special
(»
}
"n n=l
{~
be a sequence of
random variables, centered at expectations and with uniformly bounded fifth
moments.
Suppose that
E(L
(2.16)
l;n)2
= N + O(N14 / l5 ) .
n~N
Let
(2.17)
K
be a posi tivc constant.
Vie shall assume t:hat
Ip(AB) - P(A)P(B)I ~ a(n t-
K
)
-17for all
zero.
A € F~
t n
B E F + .
and all
t+n
We call sequences
{~n}
Here
a(s)
converges monotonically to
satisfying (2.17) retarded weakly
~ixing.
This condition is less restrictive than the concept of (*)-mixine, introduced
by Blum, Hanson and Koopnans (1963) because of the retardation factor
t
-K
and because we do not require that the right-hand side of (2.17) contains the
factor
peA)
PCB) .
Of course, (2.17) is also less restrictive than what we
called in (2.6) a retarded strong mixing condition since we require less con-
H: n } .
cerning the future of the process
Moreover, we assume that for all
(2.18)
i
~
j < m~ n
IE(~.~.~ ~ ) - E(~.~.)E(~ ~ ) I ~ { a((m _ j)j-K) }
1 J mn
1 J
mn
l/S
•
Of course, if we would assume (2.6), instead of (2.17), this estimate would
immediately follow from a
THEOREM 6: Let
{~n}
well~known
lemma of Ibragimov (1962).
be a sequence of random variables, centered at expec-
tations, with uniformly bounded fifth moments, and satisfying (2.16), (2.17),
and (2.18) with
K
= 2/19
and
a (s ) c: s
-16802
Then, without changing the distribution of
the process
{S(t), t
~
O}
{S(t), t
~
O} , we can redefine
on a richer probability space together with
-18.~
standard Brownian motion {X (t) , t
O}. such that
1
2
--n
S (t) - X(t) « t
for each
Let
n
a.s.
1/294 .
<
{x }
n
denote a Gaussian sequence centered at expectations.
Theorems
A - D are well-known for the case of stationary uncorrelated (that is, independent identically distributed)
As a matter of fact, for this special
xn
case relation (1.2) is rather trivial to prove.
Here we consider Gaussian sequences whose n-th partial sums have variances
close to
n
and that have covariances converging to zero as the distance
between indices approaches infinity.
In this more general setting the proof
of (1.2) is far from being trivial.
We assume that uniformly in m
m+n
>
E. ( l<=m+1
(2.19)
for some
C1
=0
.
e: > 0
xk
)2
and some constant
= a
a
n + 0 (n
2
> 0 ,
l-e
)
excluding"
the degenerate case
Hence without loss of generality we assume
a
(2.20)
2
=1
Moreover, we assume that uniformly in
(2.21)
2
E(x
x
m m+n
.
ill
-2
) = p (m, m + n) «n '
2
Since by (2.19) the variances EX k are uniformly bounded, condition (2.21) is
less restrictive than the requirement that the correlations converge to zero
-19-
at the given rate.
THEOREM 7:
l~ithout
changing the
redefine the process
~
{SCt), t
with standard Brownian
~otion
~istribution
O}
for each
T) <
COROLLARY:
{Set),
t
~ o} ,
we can
on a richer probability space
~
{XCt), t
~Ct) - X(t)
of
such that
O}
1
2
to~ether
-T)
«
t
a.s.
min(1/60, 4£/15) .
Let
{x}
n
be
stationary Gaussian sequence, centered at exPec-
8
tations and with covariances
The conclusion of Theorem 7
holes with
1
~(t)
- XCt) « t
1
260
a.s . .
Similar results are obtained for functionals of certain :1arkov processes
and for the, Shannon-McHillan-Breimann theorem in. information theory.
3.
DESCRIPTION OF THE METHOD
-'---._-Let
uniformly hounded.
be a weak! y ·--1.ependent sequence \<!i th
Pere
0
>
0
is fixed.
1
li8
n-+oo
var(n 2 Sn )
=02
and
For ease of explanation suppose
that
(3.1)
Ex = ()
n
>
0 .
-20-
Recall that
was defined in (1.1) as the n-th partial sum.
Sn
of generality we assume that
a
2
=1 .
'~'ithout
loss
As already explgined in Section lour
goal is to establish the fundamental relation (1.2) for various weakly dependent sequences of random variables.
main steps.
The proof of (1. 2) is accOn1l'liS 11ed. in t,·'o
In Section 3.1 we describe the first step and in Section 3.2 the
second step.
3.1. The first step consists of approximating {Sn} by a martingale.
~he
following simple le!'1ma concerning such approximations of sums of a.rbitrary
random variables by J'lartingales is useful.
IX>
LEMMA 1: Let {Yj}j=l be an arbitrary sequence of randoM variables and let
IX>
{L.}. 0 be a nondecreasing sequence of a-fields such that
J J=
measurable.
LO denotes the trivial a-Helc.)
(Here
y.
J
~uppose
is L.J
that the series
00
l k=O
(3.2)
for each
where
j
~
1 .
IX>
{Y.,L.}. 1
J
J J=
kIL·)1
BIE(y.
J+
Then for each
is a
j
Martin~ale
-
<
00
J
~
a.s.
1
difference sequence ane
00
U.
J
Indeed, the
desirc~
= Lk=O
P(y. kIL'_I) .
J +,
J
ma.rtingale difference sequence is given by
00
Y.
J
= Ik=O
(E(y·+k IL .) - F(y. ~IL. 1)) • j ~ 1
J.
J
J+...
J-
-21thus proving.
the~ ilenuna'.,
.~
f or th'
-' l- 'II- a t 1eas t t 0 .Ct
t l'Jevlclus
. v.
Tn
,e 1·.ea
,IS 1el'1ma can b e t rClcel!.,ac..
, au
(1969) anG Gordin (1969).
Both autllors gave a martinq-a1e representation of
certain strict sense stationary sequences,
Gor~in
in terms of unitary operators
on Hilbert space.
The significance of the lemma obviously lies in the fact that the partial
sums of any sequence of random variables satisfying (3.2) equal a martingale
plus a telescoping sum which under certain restrictions can be discarde0.
!bwever, in most cases LemMa 1 turns out to be a rather weak tool when applied
directly to the given sequence of random
vari~c1es.
It only then gains in
strength when cOTl]bined with another essential inp:redient of the l'!ethod which
we describe now.
y.
Pe define new random variables
larger blocks of the
~iven
xn
of blocks of the
~iven
xn
y. .
J
As a typical
exa~le,
J
which are sums of proeressively
Further, we soneth'1es have to construct sums
s~a1ler
than the
corresponGin~
blocks
definin~
consider the construction of the blocks
required for the analysis of t 1:e Gaussian random variables of 'l'heorel'1 E.
There we define the
y.
consecutive
x
z.
J
n
inductively by accin1
J
respectively, leaving no r:aps between the blocks.
The
z.
J
are defined in such a way that they
provide enough separation between consecutive
each
j
~
I.
and
For then, by Lemma 1, ,
y.' s
J
so that
(3.2) hold.s for
-22-
where
{Y. ,L. }
J
is a
J
~arti ngale
generated by
Poreover, and essential to the subsequent
are providing between consecutive
Zj
is so large that even
J
(3.3)
u.
J
y.
is sman compared to
= l k=O
E(y. kiL. 1)
for large
J
J+
J-
j
For then
.
y ... Y.
J
J
(,3.4)
for large
j
Lastly, since the blocys defining the
r·~
L
(3.5)
for large M
l
Yj ~
(y. + z.)
In other words the
z.
J'
u.
J
the
SN'
Of course, we are deliberately
is small comparee with
In the construction of the
definin~
a.s.
can be discarded "Tithout affecting the
J
almost sure behaviour of the partial sur.Js
If the blocks
are muc" sr.'aller
H
J
vague when saying that
J
J
j=l
j=l
Z.
y. , we have
than the blocks defining
hold.
is the a-field
J
analysis, the separation that the
y.'s
L.
JlffeL"ence sequence and
z.
J
y.
J
and
z.
J
y. , etc.
J
there is a trade-off involved:
are too small, (.3.;'.) and (3.4) ",ill fail to
However, if these blocks are too large, then (3.5) will fail to hold.
That is, the
z.'s could not be neglected in the subsequent analysis.
J
Assuming that the blocks have been chosen so that (3.2), (3.4),and (3.5)
are satisfied, we then concentrate on the sums
L
j:s:M
y.
J
-23-
in the renainder of the discussion.
Fm'1ever, consideration of these sums will
produce fluctuation results only for a certain sUQsequence of the sequence
{SN}
of partial sums.
blocks defining the
for
{SN} ,
Fortunately, it is relatively easy to break into t'1e
y. 's
]
and thus recover
t~e
desired fluctuation results
provided that tll.e blocks defining the
are not chosen too
y. 's
J
large.
By the above arguments, noting (3.4), the proof of
t~e
fundamental rela-
tion (1.2) is reduced by the first step of the method to provinr a corresponding almost sure invariance principle for the
Y.
(3.6)
3.2.
~artin~ale
J
The second step of the method is the
(3.6) in an appropriate manner by Brownian
neans of a martingale version of the
~trassen
(1965), Theorem 4.3).
approxi~ation
~otion.
~ko:rokhod
P~plication
of the
This is accomplished hy
representation theorem (see
of this representation
provides (on a new probability space possibly) stopping times
Bro~~ian
(3.7)
(3.8)
and
(3.9)
motion {X(t)} such that
I
j~M
Y.
]
=
xU
j~H
T.)
J
~artinga1e
a.s. ,
{T. }
J
theore~
for
-24-
for each
Let
[t]
p
> I
~11
.
y.
denote the index of the
inte~er
denote, as usual, the greatest
z.
or
J
J
which contains
not exceeding
t
and let
N
In order to
X
exploit (3.7) we establish a strong la,,! of large nUMoers for the T.
J
in the
form
I
(3.10)
~
j=l
where
11
1
>
0 .
T.
J
=N
+O(N1-nl)
For then we obtain for each
a.s.
11
1
2 < znl ' omitting some ca1cu-
lations and using (3.10),
X[I;:I
Tj )
(3.11)
~ X[N
+
N ~ t < N + 1.
+;
= X(N)
+
= X(t)
+ 0 t-
2
n
)
f }-n,]-
c·
for
o[NI-nIl)
a.s.
Thus if we set
S*(t)
=l
H[t]
Y.
j=l
J
we obtain, using (3.7) and (3.11),
1
S*(t) - X(t) «
(3.12)
as
t
-+
GO
"2 11 2 a.s.
t
•
Now we are almost done since, in view of (3.4), the reduction to the rnartingale case described in Section 3.1
results in the estinate
-25S (t) - S* (t)
=
rn::;t
x
n
f~*(t)
-
(3.13)
1
-··-n
Y.)«t2 .3
(y.
J
for some
n
3
>
J
a.s.
We combine (3.12) and (3.13) and get
O.
1
Set) - XCt) «t
for some
n
> 0 ,
-n
2
a.s:
which is the desired fundamental relation (1.2).
~ethod.
completes the second step of the
How can we get (3.10) which is, as we just
lishing the fundamental relation (1.2)?
r.~
-
L
j=l
~ave
r~
T. - N =
J
(3.14)
rj=l
+
(T.
E (T·I L. 1))
J J-
J
L
j=1
~~
+
L
Here we have used (3.8).
MN
r
j=l
far some
which,
j
-
J
N.
We first establish a
stron~
law of large numbers for
in the form
(3.15)
(3.14).
- y~)
(F (Y?; IL. 1)
J J?
y~
j=1
Y:J?
seen, the key to esteb-
Consider the decomposition
f/li
the
TI~is
Y~ = N
+
1 n4
O[N - )
a.s.
J
n > O. Ynis takes care of the third sum on the right-hand side of
4
2 0
As a by-product of the proof of (3.15) we obtain a bound on E!Y.1 +
1
in view of
,
1~
(3.9), yi~lds a bound on E T.
J
J
Using these two bounds
-26-
and a standard martingale convergence theorem it is not difficult to show that
l-n ~
the first two sums on the riR'1.t-hand side of (3.14) are «N
almost
v
surely, proving (3.10).
Lj •
(In general,
T.
J
is not measurable with respect to
But this is only a technicality which \'1e shall not worry about at this
point.)
For general martingales, (3.15) does not
~old.
But using.(3.. l4) it folloNs
tnat itt' is .enoughto prove;
M
N
l j=l
(3.16)
for some
n6 > O.
2
Yj
=N +
( I-n )
CN 6
a.s.
In view of the weak dependence of the
(3.16) might appear to be straight-forward.
provides the most difficulty.
Yj'S, relation
But typically, proving (3.16)
-27-
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