Local maxima of the sample functions of the two-parameter Wiener process

Tran, Lanh Tat

doi:10.1090/S0002-9939-1976-0420811-2

Local maxima of the sample functions of the two-parameter Wiener process
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by Lanh Tat Tran PDF

Proc. Amer. Math. Soc. 58 (1976), 250-254 Request permission

Abstract:

It is shown that for almost all sample functions of the twoparameter Wiener process, the set of local maxima is dense in $R_ + ^2$.

References

N. N. Čencov, Wiener random fields depending on several parameters, Dokl. Akad. Nauk SSSR (N.S.) 106 (1956), 607–609 (Russian). MR 0077824
Jean Delporte, Fonctions aléatoires de deux variables presque surement à échantillons continus sur un domaine rectangulaire borné, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 6 (1966), 181–205 (French). MR 212858, DOI 10.1007/BF00531803
David Freedman, Brownian motion and diffusion, Holden-Day, San Francisco, Calif.-Cambridge-Amsterdam, 1971. MR 0297016
Tosio Kitagawa, Analysis of variance applied to function spaces, Mem. Fac. Sci. Kyūsyū Univ. A 6 (1951), 41–53. MR 47299
Steven Orey and William E. Pruitt, Sample functions of the $N$-parameter Wiener process, Ann. Probability 1 (1973), no. 1, 138–163. MR 346925, DOI 10.1214/aop/1176997030
Chull Park, A generalized Paley-Wiener-Zygmund integral and its applications, Proc. Amer. Math. Soc. 23 (1969), 388–400. MR 245752, DOI 10.1090/S0002-9939-1969-0245752-1
Won Joon Park, A multi-parameter Gaussian process, Ann. Math. Statist. 41 (1970), 1582–1595. MR 272044, DOI 10.1214/aoms/1177696802
J. Yeh, Wiener measure in a space of functions of two variables, Trans. Amer. Math. Soc. 95 (1960), 433–450. MR 125433, DOI 10.1090/S0002-9947-1960-0125433-1
J. Yeh, Cameron-Martin translation theorems in the Wiener space of functions of two variables, Trans. Amer. Math. Soc. 107 (1963), 409–420. MR 189138, DOI 10.1090/S0002-9947-1963-0189138-6
J. Yeh, Orthogonal developments of functionals and related theorems in the Wiener space of functions of two variables, Pacific J. Math. 13 (1963), 1427–1436. MR 159216
Grenith J. Zimmerman, Some sample function properties of the two-parameter Gaussian process, Ann. Math. Statist. 43 (1972), 1235–1246. MR 317401, DOI 10.1214/aoms/1177692475