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December, 1976 Sequential Compactness of Certain Sequences of Gaussian Random Variables with Values in $C\lbrack 0, 1 \rbrack$
Gian-Carlo Mangano
Ann. Probab. 4(6): 902-913 (December, 1976). DOI: 10.1214/aop/1176995935

Abstract

Let $(X_n(t): t \in \lbrack 0, 1 \rbrack, n \geqq 1)$ be a sequence of Gaussian processes with mean zero and continuous paths on [0, 1] a.s. Let $R_n(t, s) = EX_n(t)X_n(s)$ and suppose that $(R_n: n \geqq 1)$ is uniformly convergent, on the unit square, to a covariance function $R$. It is shown in this paper that under certain conditions the normalized sequence $(Y_n(t): t \in \lbrack 0, 1 \rbrack, n \geqq 2)$ where $Y_n(t) = (2\lg n)^{-\frac{1}{2}}X_n(t)$ is, with probability one, a sequentially compact subset of $C\lbrack 0, 1 \rbrack$ and its set of limit points coincides a.s. with the unit ball in the reproducing kernel Hilbert space generated by $R$. This is Strassen's form of the iterated logarithm in its intrinsic formulation and includes a special case studied by Oodaira in a recent paper.

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Gian-Carlo Mangano. "Sequential Compactness of Certain Sequences of Gaussian Random Variables with Values in $C\lbrack 0, 1 \rbrack$." Ann. Probab. 4 (6) 902 - 913, December, 1976. https://doi.org/10.1214/aop/1176995935

Information

Published: December, 1976
First available in Project Euclid: 19 April 2007

zbMATH: 0364.60025
MathSciNet: MR428415
Digital Object Identifier: 10.1214/aop/1176995935

Subjects:
Primary: 60F15
Secondary: 60G15

Rights: Copyright © 1976 Institute of Mathematical Statistics

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Vol.4 • No. 6 • December, 1976
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