Sample Functions of the $N$-Parameter Wiener Process

Abstract

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 $d \leqq 2N (d < 2N)$.