Open Access
February, 1973 Sample Functions of the $N$-Parameter Wiener Process
Steven Orey, William E. Pruitt
Ann. Probab. 1(1): 138-163 (February, 1973). DOI: 10.1214/aop/1176997030


Let $W^{(N)}$ denote the $N$-parameter Wiener process, that is a real-valued Gaussian process with zero means and covariance $\Pi^N_{i=1} (s_i \wedge t_i)$ where $s = \langle s_i\rangle, t = \langle t_i\rangle, s_i \geqq 0, t_i \geqq 0, i = 1, 2, \cdots N$. Then $W^{(N,d)}$ is to be the process with values in $R^d$ determined by making each component an $N$-parameter Wiener process, the components being independent. Our concern is with continuity and recurrence properties of the sample functions. In particular we give integral tests for upper functions which reduce in the case $N = d = 1$ to the integral tests of Kolmogorov, and of Chung-Erdos-Sirao. We formulate and prove precise statements of the fact that $W^{(N,d)}$ is interval recurrent (point recurrent) if and only if $d \leqq 2N (d < 2N)$.


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Steven Orey. William E. Pruitt. "Sample Functions of the $N$-Parameter Wiener Process." Ann. Probab. 1 (1) 138 - 163, February, 1973.


Published: February, 1973
First available in Project Euclid: 19 April 2007

zbMATH: 0284.60036
MathSciNet: MR346925
Digital Object Identifier: 10.1214/aop/1176997030

Primary: 60G15
Secondary: 60G17

Keywords: Gaussian processes , modulus of continuity , recurrence , upper functions

Rights: Copyright © 1973 Institute of Mathematical Statistics

Vol.1 • No. 1 • February, 1973
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