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December, 1973 The Law of the Iterated Logarithm for Gaussian Processes
Hiroshi Oodaira
Ann. Probab. 1(6): 954-967 (December, 1973). DOI: 10.1214/aop/1176996803


Strassen's law of the iterated logarithm for Brownian motion is extended to a class of Gaussian processes. Let $\{X(t), t \geqq 0\}$ be a real continuous Gaussian process with $X(0) = 0$, mean zero and continuous covariance kernel $R(s, t)$. Define a random sequence $\{f_n(t, \omega)\}$ in $C\lbrack 0, 1 \rbrack$ by $f_n(t, \omega) = X(nt, \omega)/(2R(n, n) \log \log n)^{\frac{1}{2}}$. Under certain conditions on $R$ it is shown that with probability one $\{f_n(t, \omega)\}$ is equicontinuous and the set of its limit points is the unit ball of the reproducing kernel Hilbert space with reproducing kernel $\Gamma$ determined by $R$. The result generalizes the author's earlier result (1972).


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Hiroshi Oodaira. "The Law of the Iterated Logarithm for Gaussian Processes." Ann. Probab. 1 (6) 954 - 967, December, 1973.


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

zbMATH: 0272.60025
MathSciNet: MR368122
Digital Object Identifier: 10.1214/aop/1176996803

Primary: 60F15
Secondary: 60G15

Keywords: Gaussian processes , kernel Hilbert space , reproducing , The law of the iterated logarithm

Rights: Copyright © 1973 Institute of Mathematical Statistics

Vol.1 • No. 6 • December, 1973
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