I
University of North Carolina at Chapel Hill and University of New Mexico.
This research was supported in part by the Office of Naval Research under
Contract NOOOl4-67-A-0321-0002.
2
University of North Carolina at Chapel Hill. This research was supported
in part by the U.S. Air Force Office of Scientific Research under Contract
AFOSR-68-l415.
3
University of North Carolina at Chapel Hill and Kyushu University (Japan).
This research was supported in part by the National Science Foundation under
Grant GU-2059.
A NOTE ON A 0-1 LAW FOR STATIONARY GAUSSIAN PROCESSES
by
Clifford Qua11s l , Gordon Simons 2 and Hisao Watanabe 3
Department of Statistias
Uni versity of North Caro Una at Chape Z Hi ZZ
Institute of Statistics Mimeo Series No. 798
January, 19'12
A NOTE ON A 0-1
LAW FOR STATIONARY GAUSSIAN PROCESSES
by
Clifford Qua11s l , Gordon Simons 2 and Hisao Watanabe 3
1. Summary and Introduction. Let {X(t), t
Gaussian process with continuous sample functions
T
= N+,
tion
the set of positive integers.
r(t)
satisfies
r(O)
=1
T}
€
be a real stationary
and"
T
= (O,~)
Also assume that its covariance func-
and that
EX(t)
= O.
Let
f(t)
trary non-decreasing positive function on some time interval
T
= N+
we define the event
·e
A
=
or with
be an arbi-
[a,~).
For
A by
[X(n) > f(n)
infinitely often].
In Section 2, we show that the event
A has
0
or
1
probability,
provided only that the covariance function satisfies the mixing condition
r(n)
~
0-1
law of this type (a type of the law of the iterated logarithm) containing
0
as
n
~~.
Under a stronger mixing condition, we also give the
a test to decide between
0
and
1
for each function
f.
The proof of this
1
University of North Carolina at Chapel Hill and University of New Mexico.
This research was supported in part 'by the Office of Naval Research under
Contract N00014-67-A-032l-0002.
2
University of North Carolina at Chapel Hill. This research was supported
in part by the U.S. Air Force Office of Scientific Research under Contract
AFOSR-68-l4l5.
3
University of North Carolina at Chapel Hill and Kyushu University (Japan).
This research was supported in part by the National Science Foundation under
Grant GU-2059.
2
discrete time version
~
0-1
law with test is similar to and easier than the
results in Watanabe [6] and Qualls and Watanabe [3], [4] for continuous times.
In Section 3 for
B
= [X(t)
T
= (0,00),
we note that the event
on some sequence of
> f(t)
bility provided only that
r(t)
n
as
0
+
t
's
t
00]
+
also has
00.
+
or
0
1
proba-
An interesting difference be-
tween the discrete and continuous time cases is discussed in §3.
We note here that some of the results stated in the following sections
for stationary processes can be extended easily to certain non-stationary processes.
¢: t + ¢(t)
For a change of time
which is strictly increasing to
Y(t)
= X(¢(t»
also satisfies the
P{Y(t)
·e
2.
~
f(t)
t
0-1
+
¢
is a continuous function
the non-stationary process
00,
law:
on some sequence of
t
n
's
+
=
oo}
0 or 1.
Discrete Parameter Case.
THEOREM 1. If
PROOF.
as
00
where
r(n)
0 as
+
n
+
there exists a sequence of events
generated by
X(l), ••. , X(k),
(1)
-
P(A
~ ~) +
0
{~},
= 0 or
1.
[(X(l), ••• ,X(k»
€
the k-th belonging to the a-field
with
as
k
(cf. Halmos [2], Theorem D, page 56).
pressed as
P(A)
A is an event in the a-field generated by K(l), X(2), ..•
Since
E
k
00 then
F ]
k
+
00
~
Of course, the event
for some Borel set
Fk
of the k-dimen-
sional Euclidean space.
For
m
~
0,
let
=
and
[X(m+n) > f(n)
for infinitely many
n]
[X(n)
for infinitely many
n
>
f(n-m)
can be ex-
~
m]
3
1\ (m)
[(X(m+l), ••• ,X(m+k»
Because of stationarity, we obtain for each
Since
f
is monotone,
=
P(A 6 A(m»
(3)
Set
D
mk
A
C
A(m)
= Ip(A_~
(m»
-l.Ck
k,
and, hence,
for all
0
m and
F ].
k
€
m.
_ P(~ )P(A (m»/.
-1<
.,k
From (1), (2) and (3), we
obtain
2
Ip(A) - p (1\) I
=
P(~)P(1\ (m»/
D + Ip(A) - P(1\1\ (m»1
mk
$
D + P(A 6 ~) + P(A 6 1\ (m»
mk
D + E + P(A 6 A(m» + p(A(m) 6 ~ (m»
mk
k
-1<
$
$
Dmk + 2Ek •
$
But the assumption
Ip(A) -
=0
limn-+oo r(n)
implies
limm+oo Dmk
=0
(as we shall
indicate below) and, consequently,
(4)
Ip(A) -
p2(~)1
$
This can be seen by writing
2k-fold integrals (over
2E
k
for each
P(~~ (m»
F x F )
k
k
and
k.
P(~)P(~ (m»
m ~ k)
(when
as
and observing that the integrand of the
first integral, a multivariate normal density, converges, as
m~
00,
to the
integrand of the second integral, also a multivariate normal density (which
does not depend on
m).
The convergence of the first integral to the second
then follows from a version of Scheffe's theorem (cf., [1], page 224).
Finally, using (1) and letting
o and, hence,
P(A)
=0
or 1
k ~
00
as claimed.
in (4), we obtain
P(A) _ P 2 (A)
=
4
Remark.
4It
An ergodic theoretic interpretation of this proof is possible.
One considers the probability space induced by the random vector
(X(l), X(2), ••• )
"point"
and lets
(X(l), X(2), ... )
assumption makes
~, A(m)
T
one finds that
into the point
The stationary
The events
A,
can be readily interpreted in this new framework and
~ (m) = T-m~.
A(m) = T-mA,
ment as that used to show
"mixing".
(X(2), X(3), ... ).
a "measure preserving transformation".
~ (m)
and
T be the "shift transformation" which maps the
lim
m+oo
D
mk
= 0,
From this, it follows that
T
With essentially the same arguone can verify that
is "ergodic".
to Renyi [5], page 144, for a proof of this fact.
concepts used in this remark.)
Finally, since
P(A6T- 1A)
to an "invariant set" (Le.,
= 0)
T
is
(The reader is referred
The nearby text defines the
A is almost surely equivalent
it follows that
=0
P(A)
or 1.
THEOREM 2.
-4It
(i)
If I(f) -
(ii)
Since
assumption
lemma yields
r(n) = O(n- Y)
and
= X(k)
~
Denote
Part (;).
00
n
as
<
00,
-+
00
,
P(A) = O.
then
for some
then
Y > 0,
= 1.
P(A)
PROOF.
=
If I(f)
(f(k»-l exp(-f 2 (k)/2)
I~=k 0
P(~
I(f) <
00
and
> x)
makes
~
= f(k).
fk
(2n)
-~
x
-1
2
exp(-x /2)
I~=l P(~ > f ) <
k
x > 0,
for any
the
Then the Bore1-Cante11i
00.
P(A) = O.
Before beginning the proof of part (ii), we introduce the following
lemmas.
LEMMA 1.
If (ii) is true with the additional restriation
2 log log k
for all
PROOF.
~k
Set
k
~
some k O'
= max(fk'~)
where
f
2
k
~
2 log k -
then it is true without the restnation.
1<-
~
". ".
= (2 log k - 2 log log k)2 (k>1);
additional restriction does not hold, then
~k
=
~
infinitely often.
If
Since
the
5
[~> ~k infinitely often]
the event
~
But letting
~
k
it suffices to show
~k
along a subsequence for which
00
00
I (~)
A,
c
j=k
j=k
O ~
-1
exp(-~
2
00.
we find that
~ j -1 exp(- ~ j 2 /2)
L
~
0
(k-k )
~
k
~ j -1 exp(- ~ j 2 /2)
L
=
= uk'
I(~) =
0
/2)
=
1.::
~
00.
(2 log k - 2 loglog k)2 • k
That is,
I(~) =
LH1~4A
Let
2.
Ir(n)1 <
PROOF.
£
be given.
> 0
n ~ l~
for
£
00
If (ii) is true with the additionaZ restriction
then it is true without the restriction.
Suppose the restriction does not hold.
Since
r(n) = O(n- Y)
we have
~ - X·
(n ~ 1) is a stan
mn
tionary Gaussian sequence with covariance function ~(n) :: r(mn) satisfying
for some
~(n)
m,
= O(n- Y)
often]
c A,
Ir(n)1 <
and
1~(n)1 <
£.
Defining Ek
=
[~ ~ f
k ],
m~i<j~n
<p(x,y; Ar)
Zation coefficient
PROOF.
I(f)
=
Since
00
entails
[~n
>
I(h
~
n
=
infinitely
00.
But this
f •
n
~ L L
where
Now
~ n :: f mn
Let
it suffices to show that
follows from the monotonicity of
LEMMA 3.
n ~ m.
whenever
£
we have (for
1 ~ m < n)
Ir(j-i)I fl <P<f.,f j ; Ar(j-i»
0
~
dA
denotes the standard nOY'l7laZ bivaY'iate density with cOY'Y'eAr.
This type of lemma appears in many proofs of asymptotic independence
for crossing problems.
Watanabe [3].
Lemma 3 is a special case of Lemma 1.5 in Qualls and
6
Proof of Part (ii). The assumption l(f) =
hence,
IT
00
1
(5)
peEk) = O.
1 - peA)
=
00
L~ P(E
implies
k
c) =
00
and,
In turn, we have
lim
m-rOO
P(~kJ
m
00
=
=
lim
m -r 00
{P(~k]
m
In view of Lemma 3, we may conclude
peA) = 1
providing
Now assume the restrictions in Lemmas 1 and 2 hold with
For
Ir I
<
£ = min(1/4, y/5).
£,
2
2
constant· exp{-(l-lrl)(fi +f )/2}
j
constant· exp{-(l-£)(f 2+f 2 )/2}
j
i
Then
constant.
1
f
f
k- 5 £ ~(10g i) (lOgCi+k) 11 -£
i=l k=l
~ i J i +k U
00
constant •
00
L L
i=l k=l
3. Continuous Parameter Case.
The continuous time version of Theorem 1
depends on the sample path continuity of
X(t).
Specifically, the values of
7
X(t)
~
on
(0,00)
n
{k/2 , k and n
generated by
erated by
are determined by the values on the diadic rationals
N+}
€
and hence the event
= k/2 n ,
{X(t), t
= k/2 n ,
{X(t), t
k
k and n
€
F
B is a member of the a-field
+
N }.
= 1, ••. ,n2 n }
Fn be the a-field gen-
Let
(n ~
1).
The sequence
Fn is
OO
F is the smallest a-field containing the field U1 Fn·
non-decreasing and
Using Halmos' result as we did in proving Theorem 1, we may obtain events
Bk
Fk for which Ek
€
Bk
is a set
Fk
= {(X(1/2 k ),
Bk
= P(B
~ B ) + 0
as
k
k
+
00.
Associated with each event
k
k2 -dimensional Euclidean space with
in
..• , X(k2 k /2 k »
F }.
k
€
By imitating
the remaining steps in
the proof of Theorem 1, we obtain:
THEOREM 3. If r(t)
.-
0
+
as
t + oo~
then
P(B) = 0 or 1.
Remark. However, the similarity of Theorem 3 to Theorem 1 does not extend to
the
law with test.
0-1
For example, in [3], the continuous time version of
our present Theorem 2 not only assumes the mixing condition
t
00
+
for some
r(t)
=
Y
0
>
1 - cltl
a
as
There the test is based on whether
local condition.)
sum
fooa
I(f)
= O(t- Y)
as
but also a local condition:
+ o(ltl a )
is finite or infinite.
r(t)
t
+
J(f)
0,
where
= fooa
0 < a ~ 2
and
C
>
o.
{f(t)}2/a-l exp(-f 2 (t)/2) dt
(Reference [4] gives a test for a somewhat more general
Since
x-I exp(-x 2 /2)
is a monotone function of
x,
the
in Theorem 2 may be replaced by the integral
{f(t)}-l exp(-f 2 (t)/2) dt, which is less than the integral
continuous time case (when
are functions
f(t)
a
such that
is chosen sufficiently large).
P(B)
=1
and
P(A)
= O.
J(f)
for the
In fact, there
8
References
[ 1]
Billings ley, P. Convergenoe of Probabi U ty Measures.
Wiley, New York, 1968.
[2]
Halmos, P.R.
[3]
Qualls, C. and Watanabe, H. "An asymptotic 0-1 behavior of Gaussian
processes." Ann. Math. Statist. 42 (1971), 2029-2035.
[4]
Qualls, C. and Watanabe, H. "Asymptotic properties of Gaussian •
processes." University of North CaroUna Institute of Statistios
~meo Series No. 736 , Chapel Hill (1971).
[5]
Renyi, A.
[6]
Watanabe, H.
Measure Theoy.y.
D. Van Nostrand, Princeton, 1950.
Foundation of ProbabiUty.
Holden-Day, San Francisco, 1970.
"An asymptotic property of Gaussian processes."
Trans. AMS 148 (1970), 233-248.