* This work
\dll appear as Chapter 12 of Philipp and Stout (l974a).
** This \'lOrl: was partially supported by the !.Jational Science Foundation and
partially by Naval Contract nOOOl4-67-A-032l-002 TSK NP.-042-214.
12.* ALMOST SURE INVARIANCE PRINCIPLES FOR
CONTINUOUS PARAMETER STOCHASTIC PROCESSES. **
by
Nal ter Philipp
university of Illinois at Champaign-Urbana
and
William F. Stout
University of IUinois at Champaign-Urbana
and
University of North Carolina at Chapel HiU
Insti tute of Statistics !fimeo Series No. 9fi4
November 1974
THIS PUBLICATION
STATISTICS HIMEO
t1EAKLY DEPENDENT
DATED AND HAILED
IS TO BE CONSIDERED IN CONJUNCTION WITH WE INSTITUTE OF
SERIES NO. 950 - "ASYr,~PTOTIC FLUCTUATION BEHAVIOR OF SUMS OF
RANOOH VARIABLES" BY WALTER PHILIPP AND tJILLIAM F. STOtIT TO YOU SEPTEHBER, 1974.
12.* A1mo2; Sure Invaripn~~ Principles for
Continuous Parameter Stochastic Processes. **
by
T'lalter P!'ilirm
University of Illinois at
Champaign~Urbana
and
Pilliam F. Stout
University of Illinois at Champaign-Urbana
and
Department of Statistics
University of North Carolina at ChapeZ HilZ
Introduction
12.1
~tout
In Philipp and
(1974a) almost sure invariance principles are estar.-
lished for certain discrete parameter stochastic processes.
marized in Philipp and Stout (1974t).
This work is sum-
Tre results of Philipp and Stout (1074a)
can be extended in a stra"ight:C'orward fashion to continuous parameter stochastic
processes.
Here we illustrate this on processes ,,,hose increments are either
jointly Gaussian or </>-·mixing.
Let
usual,
{2 (t), t
~
{JC(t), t ~ O}
O}
denote the process und.er consic1eration and let, as
denote standard Brownian motion.
1-'e sh:?l1 establish the
*------This
wor~
\;Jill appear as Chapter 12 of P!,ilipp and Stout (1974a).
** Tt.is wor!: was partially supported. by the National Science Foundation and
partially by 'NavaJ
GontractN00014-67-A-O~21-002
""'SK
N~-04?-21·L
-2almost sure invariance principle
1
Set) - X(t) « t
as
t
+
for some
00
n >
n.
-n
2
a.s.
Since, usually
1
S([t)) - X(t)« t
--n
2
a.s.
(12.1.1)
is an immediate consequence of a discrete parameter almost sure invariance
principle it remains to show, by means of an appropriate p"axil"lal inequality,
that
1
Set) - S([t])« t
as
12.2
t
+
00
--n
2
a.s.
(12.1. 2)
•
Pt'ocesses Hith
~at.!ssi:ln
IncreP1ents
Throughout this section we assume that
{S(t), t
~
O}
is a separable
Gaussian process centered at expectations with continuous covariance f'unction
~(s,t)
and
S(O)
=0
Theoren 12.1:
= E{S(s)S(t)}
.
Suppose that, uniformly in
s,
(12.2.1)
as
t +
00
for some
£ >
0 .
further loss of generality.
Suppose that
a
2
>
0 , assuminp
a
Voreover, assume that, uniformly in
2
=1
s
without
-3-
E{(S(s
as
t
~ ~.
ljJ
+ 1) -
S(s
+ t))(S(s + 1) - S(S))}«t-
2
(12.2.2)
Finally, suppose that
E{S (s
where
+ t
S(s)}2 s: ljJ2 (t) ,
+ t) -
0
$
t
$
1,
5
~ 0
is an increasing function satisfying
(12.2.3)
for some integer p > 1.
process
Then without changing the distribution of the
{S(t)}, we can redefine
{Set)}
together with standard Brownian motion
on a richer probability space
{X(t)}
such that
1
Set) - XCt) «t
as
t
~ ~
for each
p (c
.,
, 0)
n
2- n
a.s.
< min(1/60, 4e/15) .
is differentiable, then the hypothesis (12.2.2) may be
replaced by the hypothesis that, uniformly in
a2
---~R(u.v)1
auov
s,s+t «t
as
t
~ ~
Proof:
s,
-2
(12.2.2a)
.
By (12.2.1) and (12.2.2) it follows from Theorem 5.1 of Philipp and
Stout (1974a) (Theorem 7 of Philipp and Stout (l974h)) that (12.1.1) holds on
some probability space for each
{S([t]), Set)}
spaces.
and
By choosing
{Set),
{f' (t)}
n < min(1/60, 4e/15) .
X(t)}
and
Note that
are perhaps defined on different probability
{X (t)}
concli tionally independent given
-4-
{S ([t])}
on still another probability space the j oint distributions of {F' ([t]) }
and {X(t)} as well as of {S( [tJ)} ann {: (t)} are preserved.
By Fernique1s lemma (1964) (see also fbrcus (1972))
1
P{SIlPn_<t<,_n +J_ Is (t)
- S (n)
I
>
n4)
f'l
«
n
which implies (12.1.2) in view
Corollary 12.1.
Let
{x(t)}
centered at expectations.
t~at
Suppose
r
a2 = 1
(-~:?)dU «
4-
the Borel Cantelli lemlI1.a.
0'"
1Je a ~easurable stationary Gaussian process,
t~e
Denote
covariance function by
2
~~~.
2
f:
r(t)dt >
t +
00
T~reover.
without further loss of generality.
U,: (t), t
•
~ O}
S (t) =
Then, without
process
motion
chang~_ng
{f(t)}
{X(t)}
J:
x (s)ds
the distribution of
{Set), t
~
0}
l-'e can redefine the
on a ric"cr proh a 1"ility S'1ace together witt- standard Brm-mian
sud: t:lat
Set) - X(t) « t
00
suppose that
by
1
t +
= r(s,t)
(12.2.4)
-n
as
ret - 5)
a ,
ret) « t -2~
as
exp (-n~) •
is integrable and that
0
.
assumIng
exp
1
for each
n < 1/60 .
2
a.s.
-5This is an
i~mediate
E{S(s + t) - S(s)}2
=2
consequence
Ito fV
o~
Theorem 12.1.
Indeed, hy (12.2.4)
r(u)duQV
0
= 2t Joo r(u)du
o
=t
establishing (12.2.1).
+
1):- S(s
rr
o
t
r(u)dudv
v
Similarly, using the stationarity of
It JV
+ t))
o
= r(0)
dudv
c
t
2
= ¢2(t)
{x(t)}
,
say.
0
Finally, using (12.2.4)
Hence (12.2.3) is satisfied.
+ t
foo r(u)cu - 2
+ D(log t)
E{S(s + t) - S(s)}2 s 2r(O)
IE{(S(s
- 2t
(8(s
+
1) - S(s))}!
-s+l fS+t+l
~ J
s
s+t
In
I
x(u)x(v) dudv« t
-2
.
This proves (12.2.2) and. t'-:us the corollary.
Oodaira (1973) shows that the integral
Set) =
of the Ornstein-!JhlenbecY. process
the iterated logarithm.
function
ret - s)
12.1 that for
(T1"e
(12.2.5)
x(s)ds
{x("!:)}
obeys ftrassen' 5 functional 1a,,' of
~rnstein-l'hlenhecY
=h
ex~(-.y(t
2
{S (t)}
J:
- s))
for
process has covariance
S :S t
.)
It follows from Corollary
aI', alr.1ost sure invariance nrinciple holds.
paper Oodaira considers sone more eX8r.>ples of
obeys the functional laid of
t~e
x(t)
I'Those inte,l!.ral
In his
Set)
iterated logarithm, sue', as processes wi tr.
-1)-
real numbers, (the spectral density of the solution to the second order stochast
differential equation
the Ornstein-Uhlenbeck process being a special case).
Corollary 12.1 shows
that an almost sure invariance principle holds for the corresponding processes
S (t)
It should. be remarked that various authors ohtain laws of the iterated
logarithm and related upper and lower class results for stationary Gaussian
processes
{x(t), t
~
O}
(as contrasted with our study of
Set)
given by
(12.2.S). 2ee the papers by C'odaira (1972), (l973), by Lai (1973) and by Pathak
and Qualls (1973).
12.2 Processes with MixinQ Incrcnerts
~!e
will consider only processes with <p-mixing increments, since i t will he
obvious how to handle other kinds of mixing.
let
FOT
a given stochastic process,
b
Fa be the a-field generated by {Set) - Sea), a ~ t ~
{Set), t
~
o}
is said to have <p-mixing increments if there exists a non-
decreasing function
oo
B E Fs+t
Then
h} .
<P(t)
~
0
such that for
s
~
0 ,
t
~
0 ,
A
E
~s
f)
,
we have
Ip(AB) - P(A)P(B)
)'Ie assume throughout that
{Set)}
I
~ <P(t)P(A) .
has <p-rnixing increflents with
(12.3.1)
-71
(
We will say that
{Set)}
~
{E(s + t) - 5(s). t
throughout that
(12.3.2)
00
has strictly stationary increments if the process
O}
sa~e
has the
= '1,
S (0)
</>2(t)dt <
55 (t)
~ SUPO~t~l
p
distrihution for each
= 0,
2 0
I~(
.~ t ) 1 +
~
t
<
00
0
SOrl~
and tr.at for
0
~
s
Suppose
c5
> 0
(12.3.3)
•
Now (12.3.2) is easily seen to imply the existence of the limit
(12.3.4)
~e
(see Lemma 4.2.2 of Philipp mld Stout (1974a)).
0
2
assume throughout that
2
0=1
so that without loss of generality we can assume that
> 0
Theorer 12.2.
{Set)}
Let
be a separable process, with </>-mixing and strictly
stationary increments, centered at expectations and satisfying (12 ..3.2), (12. ,.3)
and (12.3.4) with
0
2
= 1.
Then, on an aIJpropriate probability space,
1
Set) - Z(t) «
as
t +
Proof:
Stout
00
for each
n
<
0/(12 +
~o)
n
t 2a.5.
.
(~heorem
By TheoreR 4.1 of. Philipp and Stout (1074a)
(197~b)),.relation
(12.1.1) holds on some prohahility space.
and. stationarity imply that
sUP()~t~l l~(t) I and.
have the same distribution.
Thus by (12.,.3)
sUPn~t~n+l
1
p{supn~ t
~n+
2 of Pl'1ilipp and
1 !S(t) - Sen)
I
>
n
2
--n}
{
P SU?O~t~l
=
~ E~ 5uPO~t~1
IS(t)l>
I '~ ( t ) 1 +
2 0
n
n
Separability
IS(t)· - Sen)
I
~~n}
..
- (2+0)
f'~n)
.
.2 ~
-1-0/3
«n'
-8-
(12.1. 2) follows now from the Borel-Cantelli 1em.ma.
{x(t)}
For a given stochastic nrocess
ty
{x(t). a ~ t ~ b}.
T)",en
a nonincreasinp: function
and
B
{xCt)}
Ht) {- 0
is said to rye
sue', that for?-l1
P(A)P(B)
{x(t)}
~-mixing
i f there exists
s ~ 0,
t ~ 0,
t
J
0 > 0
is a fixed nUF1ber.
f(t)
It is easy to see that if
increments with the same
A
€
G~
~ ~(t)P(A)
2 0
+
rls
<
00
=
J:
(12.3.5)
•
(12.3.6)
x(s)ds
is a ¢-mixing process,
Let
expectations and
~-mixing satisfyin~ (12.~.2)
{x(t)}
c:~uppose
~e
~2 -_ 1
and (12.3.S).
in (17.3.4).
v
1
Set) - X(t) « t
for some
n > 0 .
has
~-mixing
strictly stationary, measurahle, centered at
probaLility space
00
{Set)}
.
Corollary 12.2.
defined by (12.3.6).
t > 0
!!Trite
{xes)}
~
I
is measurable and that for all
o p,lx(s) I
t +
the a-field ~enerated
CXJ
!'Je assume throughout that
as
n"3.b ~e
G
s+t
E
Ip(AB) -
Here
let
TLe 6.eore!'1 is uroved.
i- n
a.s.
T~e~
Let
Set)
be
on an appropriate
-9-
This corollary is an immediate consequence of 'T'heorem 12.2, noting that
(12.3.5) implies (12.3.3).
1"'- ..c,.
,~.
.ar;~ov
Processes "a
<: t'
'
-," ,
.15 fy1nq
Lioe."1'1n I s c'"cnuHHV"
1'\
As an application of the results of the previous section we briefly discuss
functiona1s of Markov processes satisfying Doeblin's condition.
an abstract state space, where
Let
C'
B is the a-field of subsets of .....
stationary transition probabilities ror a Markov urocess definec on
t.inder the Doeblin condi tioD., the Markov transition function
XES,
B E
f:toc~astic
B)
h<l.s a unique stationary distribution
7T.
(S s B)
Assume
(8,B) .
(p (t, x, B), t
~ 0
~~oreover (see Doob,
Processes" p. 256) convergence to this distribution is exponentially
fast:
Ip(t,x,B) - 7T(B) , ~ ypt
for 'some
mind,
{x(t)}
y" > 0', 'P
we
'<' 1 ". uniformly
such that
(12,4.1) holds,
Let
Let
XES
and
B
€
B.
With this' in
define a Doeblin process to be a stationary f·Aarkov process
j')
bl'In proee,55 e s are "
'..
...oe·
'I'-I11Xlng
(5,8).
in
(12.4.1)
f
{y(t)}
'1"h
111_,
It is easily shmm that discrete parameter
'" (n.._,) ::: r~,p,
n
p < 1 .
'V
be a Doeblin rrocess ",ri th abstract state space
be a real valued function defined on
(S,B) ,
Suppose that
··10-
is measurable with
E x(O)
for some
0
>
0.
=0
Elx(0) 12+ 0 < ~
and
Let
_ Jrt
Set) define
{Set)} .
(12.3.4).
(12.4.2)
Su!'pose that
o
2
(j
'=
x(s)ds
1
in
Then, without changing the distribution of the process
can define
{Set)}
{8(t)}, lc7e
on a richer probability snace together with stan<1ard
Brownian motion such that
Set) holds for some
~~(t)
1
2
-n
«t
a.s.
(t -+ (X»)
(12.4.3)
n > 0
This theorem is an imJTIediate consequence of Corollary 12.2.
PJ::FEREI'IC!="$
-----Fernique, X.
(1964).
Continuite des processus gaussiens.
C. R. Acad. Sci.,
Paris 258, 6058-6000.
Lai, Tze Leung
(1973).
detection proh1ems.
Lai, Tze
LeL~g
(1~74).
Gaussian processes, moving averages and quick
Ann. Prob.:, .!.. 825-837.
Reproducing kerna1 Hilbert s?aces and the law of the
iterated logarithm for Gaussian processes.
29, 7-19.
z.
Wd1rscheinZichkeitstheorie~
-11-
I1arcus? M.
(1972) .
A bound for the distribution of to.e maximum of continuous
Gaussian processes.
Oodaira, Hiroshi
(1972).
i~>
Ann. Math. Statist"
305-309.
0n Strassen?s version of the law
o~
the iterated
Z. Wahrsoheintiohkeitstheorie,
logarithm for Gaussian processes.
~,
289-299.
Oodaira, H.iros'd.
processes.
(1973).
Ann.
Tl-e la'" of the iterated logarithm for Gaussian
Frob.~.
!.,
~5£t-067.
Pathak, P. K. and 0ualls, Clifford
(1')73).
for stati.onary Gaussian nrocesses.
Philipp, lJalter and Stout, William F.
A law of the iterated logarithm
Trans. Amel". Math. Soo., 181, J.85-193.
(l974a).
Almost sure invariance
principles for surr.s of weakly dependent random variables.
!!niversity of
Illinois Department of Hathematics preprint.
Philipp, :.Talter anc. Stout> ;'Tilliam F.
(l q74b) .
Asymptotic f1 uctuation
behaviour of sums of weakly dependent random variables.
;lo rth Cgrolina Department of Statistics)
~liT'1eo ~eries
University of
no. 950.