LOCAL MAXIMA OF THE SAMPLE WIENER PROCESS

PROCEEDINGS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 58, July 1976
LOCAL MAXIMA OF THE SAMPLE
FUNCTIONS OF THE TWO-PARAMETER
WIENER PROCESS
LANH TAT TRAN
Abstract.
It is shown that for almost all sample functions of the twoparameter Wiener process, the set of local maxima is dense in R+ .
Let {X(t):t G F2} be the two-parameter Wiener process, that is a real
valued Gaussian process with zero means and covariance
min (sx, s2 ) min (tx, t2 ),
where s = (sx,s2), t = (tx,t2) are two points of F2 . Our purpose is to show
that for almost all sample functions of A, the set of local maximal is dense in
F2 . In the one-parameter case, almost all Brownian sample functions are
monotone in no interval. As a consequence of this property, it can be shown
that for almost all sample functions, the set of local maxima is dense in F+.
A proof is given in Freedman (1971). However, this method of approach does
not provide a suitable proof in the two-parameter case due to the complex
behavior of the sample functions.
The notation of Orey and Pruitt (1973) will be used. For 5 = (sx,s2) and /
= (ii,r2) with sj < t¡, i = 1, 2, X2=,[s,, t¡] is denoted by A(s, t) and A(r) in
case s = (0,0). Let s, t G F2 . The variance of X(t) - X(s) can be verified to
be the two dimensional Lebesgue measure of S(s, t) where S(s, t) is the
symmetric difference between A(i) and A(r). We will often state that A has
continuous sample functions, independent increments and denote the increments of X over A(s, t) by A(A(s, ;)). For an account of these properties and
further information on A, consult Kitagawa (1951), Chentsov (1956), Yeh
(1960, 1963, a, b), Delporte (1966), C. Park (1969), W. J. Park (1970),
Zimmerman (1972).
Definition. X(-, co) has a local maximum at 5 if there is an open set O
containing s such that O G F2 and X(t, co) < X(s, co) for all t G O.
Lemma 1. Let s G U where U denotes the unit cube and CG U be a cube with
center at s, sides parallel to coordinate axes and equal to a. Let I, dC be the
interior and boundary of C. Then there exists a constant a > 0 and an e > 0 such
that
Received by the editors October 20, 1975.
AMS (MOS) subject classifications(1970). Primary 60G15; Secondary 60G17.
Key words and phrases. Wiener process, local maxima, independent increments.
© American Mathematical Society 1976
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250
THE TWO-PARAMETER WIENER PROCESS
251
P (sup X(t) > sup X(t) + ax/2\ > a,
Lie/
re3C
J
whenever a < e.
Proof. Let («i,m2) and (^1,^2) De tne ^east and largest vertex of C. Then
sx = (ux + vx)2 , s2 = (u2 + v2)2~x. Define
Ax ={X(sx,u2)-X(ux,u2)>2axl2),
A2 = (X(sx,u2)-X(vx,u2)>2axl2},
A3 = {X(ux,s2)-X(ux,u2)>2axl2},
A4 = {X(ux,s2) - X(ux,v2) > 2ax'2).
Now Ax, A2, A-$,A4 are independent because X has independent increments. Pick C with a < min(5,,52); then the variances of
X(sx,u2) - X(ux,u2),
^(5,^2)
- X(vx,u2),
X(ux,s2) - X(ux,u2),
X(ux,s2) - X(ux,v2)
are greater than 2~2ca, where c is any positive number smaller than sx and s2.
Therefore,
P(AXA2A3A4) = P(AX)P(A2)P(A3)P(A4)
> y
for some constant y.
Let A = AXA2A3A4. Define
(1)
M = {sup X(t) > sup X(t) + ax/2\
Líe/
íg3C
J
Let A(r,I) E U be any interval with both sides smaller than or equal to a.
Clearly,
P(M) > pÍm/A CIr sup |A-(A(r,/))|< a^A \
(2)
■p(a
n [sup|^(A(r,/))|<a'/2]).
For each t E dC, construct a point p' E I as follows: If t is not a vertex of C,
letpr be the point obtained from the perpendicular projection of / onto the line
passing through the center of C and parallel to the side of C on which / lies.
If t is a vertex of C, pick p' to be the center 5 of C. Now
P {[ mfJW)
- X(t)) > a'/2]/¿I
H [^|^(A(r,/))|
< a1/2] ] = 1.
It then follows that
p(m/A n T sup |AT(A(r,/))|< ax/2\ j = 1.
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By
(2), we obtain
252
L. T. TRAN
P(M) > PÍA n sup |A(A(r,/))| < a1'2).
\
r,lBU
/
But P(A) > y and F(supr,et/|A(A(r,/))|
< ax/2) goes to 1 by Theorem 2.1 of
Orey and Pruitt (1973). The proof of the lemma is completed.
Lemma 2. Let {C„} be a sequence of cubes with common center s, sides parallel
to coordinate axes and Cn D Cn+xfor all n > 1. Corresponding to each cube C„,
let Mn be defined as in (1). Suppose that the sides of Cn are small enough so that
P(Mn) > a for all « > 1. Let the sides of Cn be an and assume an -> 0 as n
-» oo. Define
S„ = (sup X(t) < sup X(t) + axJ2\
Lre/„
ie3C„
J
where In and dC„ are, respectively, the interior and boundary of Cn. Define
m
Em= CIS,.
///t is a constant < 1, then there exists a & > m such that
P(MkEj>tiP(Mk)P(Em).
Proof. Pick Ck with k > m and let t G F2 . Since X has independent
increments and continuous sample functions, it is possible to write X(t)
— X(s) as the sum of two independent random variables, say Hk(t) and Jk(t),
chosen in such a way that Jk(t) and Mk are independent; also Hk(t) converges
to zero almost surely as k goes to infinity. To do this, consider the symmetric
difference S(s, t) between A(s) and A(t). This symmetric difference is the union
of two parts: S(s,t) n S(tk,sk) and S(s,t) n S(tk,sk)', where ' denotes the
complement and tk, sk are the least and largest vertex of Ck. Each part can be
considered as the union of at most two nonoverlapping intervals. It is now
clear that Hk(t) can be chosen so that it depends only on the increments over
intervals that form S(s, i) D S(tk,sk); and Jk(t) can be chosen so as to depend
only on increments over intervals that form S(s, t) D [S(tk,sk)]'. Since A has
independent increments, Jk(t) is independent of Hk(t) and Mk. Pick e > 0.
Define
F=
\sMV\Hk(t)\<el.
Clearly,
P(MkEm) > P(MkEmB)
> p(Mk
\
O (sup(A,
,= 1 ^teli
> F(A4 n (sup(4(r)
\
> P(^k
'=1 Lie/,
- Xs) <
sup (X, - Xs) + fl,1/2)ß)
IE3C
J
+ Hk(t)) < sup (Jk(t) + Hk(t)) + a1/2} /b)p(B)
tsBCi
) I
A (sup Jk(t) < sup Jk(t) + ax/2 - 2e) /b)p(B).
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\
/
'=1 Líe/,
re3C,
J I
)
/
THE TWO-PARAMETER WIENER PROCESS
253
But both B and Mk are independent of
H { sup Jk(t) <
'=1 Lie/,
sup Jk(t) + ax/2 \.
ledC,
J
Therefore,
P(MkEm)
> P(Mk/B)p(
n ( sup Jk(t) < sup Jk(t) + a)12 - 2e\\p(B).
v=l
Vteii
re3C,
J J
But P(Mk) > a for all k > 1; and P(5) -+ 1 as A;-> oo since the sample
functions of X are continuous. The proof is completed by picking k sufficiently
large and e sufficiently small.
Theorem. For almost all sample functions of X, the set of local maxima is
dense in /?2 .
Proof. Let s E R+ with sx > 0, s2 > 0. By Lemmas 1 and 2, we can pick
a sequence of cubes {C„} with center at s and U D Cn D Cn+Xfor all n > 1
such that
P(Mm+xEm) > lxP(Mm+x)P(Em)
for all m > 1.
^m+i)
- P(EmMm+x)
Now
= nEmMm+x)
<P(EJ(\
= />(£j
-pa)<P(£,)(l-p«r.
Therefore />(n^'=i5m) = 0. Since í is arbitrary and C, can be chosen
arbitrarily small, the theorem follows.
Remark. Other properties related to the local maxima of Brownian sample
functions as presented in Freedman (1971) also hold in the two-parameter
case. The proof of these properties follows the same line of argument used in
the one-parameter case and therefore is not included here.
References
1.
N. N. Cencov (1956), Wiener random fields depending on several parameters,
Dokl. Akad.
Nauk SSSR 106, 607-609. (Russian) MR 17, 1101.
2. J. Delporte (1966), Fonctions aléatoires de deux variables presque sûrement à échantillons
continus sur un domaine rectangulaire borné, Z. Wahrscheinlichkeitstheorie
und Verw. Gebiete 6,
181-205.MR 35 #3723.
3.
D. Freedman (1971), Brownian motion and diffusion, Holden-Day,
San Francisco, Calif. MR
45 #6074.
4.
T. Kitagawa (1951), Analysis of variance applied to function spaces, Mem. Fac. Sei. Kyushu
Univ. Ser. A. 6, 41-53. MR 13, 855.
5.
S. Orey and W. E. Pruitt (1973), Sample functions of the A'-parameter Wiener process, Ann.
Probability 1, no. 1, 138-163.MR 49 # 11646.
6.
C. Park (1969), A generalized
Paley-Wiener-Zygmund
integral and its applications,
Proc.
Amer. Math. Soc. 23, 388^100.MR 39 #7058.
7. W. J. Park (1970), A multi-parameter Gaussian process, Ann. Math. Statist. 41, 1582-1595.
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
MR 42 #6925.
254
8.
L. T. TRAN
J. Yeh (1960), Wiener measure in a space of functions of two variables, Trans. Amer. Math.
Soc. 95, 433^50. MR 23 #A2735.
9.
-(1963a),
Cameron-Martin
translation theorems in the Wiener space of functions of two
variables,Trans. Amer. Math. Soc. 107, 409^t20. MR 32 #6565.
10.
-(1963b),
Orthogonal developments of functionals and related theorems in the Wiener
space offunctions of two variables, Pacific J. Math. 13, 1427-1436. MR 28 #2433.
11.
G. Zimmerman
(1972), Some sample function properties of the two-parameter
process,Ann. Math. Statist. 43, 1235-1246.MR 47 #5948.
Department of Mathematics,
Indiana University, Bloomington,
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