A Note on Uniform Convergence of Stochastic Processes

Zero-one laws for Gaussian processes

“…We shall show that the zero-one law of Shepp and Varberg, in fact an extension of it, follows as an immediate consequence of Theorem 1. which for each /, converges almost surely is called the orthogonal or KarhunenLoève expansion associated with the process x(/). The conditions under which (6.5) converges uniformly with respect to / almost surely have been discussed recently in the literature (see e.g., [14]). Without additional assumptions on x(/) we can deduce from Theorem 1 the following zero-one law.…”

Note relating Bochner integrals and reproducing kernels to series expansions on a Gaussian Banach space

“…By footnote 1 however flLk(X)xLk\~ í \\P\\2ÍL¡(X) < oo For applications of the Proposition see [3], [4]. In [6], Walsh applied the Chatterji Theorem [1, Theorem 1] in much the same way as here, to the Wiener measure case. The reproducing kernel representation (a) was not given however.…”