Zero-one laws for Gaussian processes

Abstract

Abstract. Some zero-one laws are proved for Gaussian processes defined on linear spaces of functions. They are generalizations of a result for Wiener measure due to R. H. Cameron and R. E. Graves. The proofs exploit an interesting relationship between a Gaussian process and its reproducing kernel Hubert space. Applications are discussed.

A Zero-One Law for Gaussian Processes

“…Introduction. For the background and history of the problem we refer to [l]and [2]. We would like to point out here that Jamison and Orey in [l] proved the above result for the special case where the Gaussian process involved had continuous paths.…”

“…We would like to point out here that Jamison and Orey in [l] proved the above result for the special case where the Gaussian process involved had continuous paths. Kallianpur [2] proved such a result for r-modules (groups closed under multiplication by rationals). Kallianpur's result for groups is restricted to those which are 73(X)-measurable rather than B0(X)-measurable.…”

“…Kallianpur's result for groups is restricted to those which are 73(X)-measurable rather than B0(X)-measurable. He points out in [2] why his proof does not work for B0(X)-measurable subgroups. Our main result (Theorem 1) unifies and generalizes the results of [l] and [2], and also gives an answer in the affirmative to the conjecture made in [l ].…”

“…He points out in [2] why his proof does not work for B0(X)-measurable subgroups. Our main result (Theorem 1) unifies and generalizes the results of [l] and [2], and also gives an answer in the affirmative to the conjecture made in [l ]. Our method of proof is similar to the one given in [2], but it is simpler.…”

Oscillation Function of a Multiparameter Gaussian Process

“…The following lemma is given in [5] as Lemma 6. Although this lemma is stated there only for a real-valued function g, the same proof works for extended real-valued functions as well.…”