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 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 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. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 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). License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use \ / '=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. 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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, License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use Indiana 47401 Gaussian