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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

A law of iterated logarithm for stationary Gaussian processes


Authors: Pramod K. Pathak and Clifford Qualls
Journal: Trans. Amer. Math. Soc. 181 (1973), 185-193
MSC: Primary 60G15; Secondary 60F20
DOI: https://doi.org/10.1090/S0002-9947-1973-0321170-8
MathSciNet review: 0321170
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Abstract: In this article the following results are established.

Theorem A. Let $ \{ X(t):0 \leqslant t < \infty \} $ be a stationary Gaussian process with continuous sample functions and $ E[X(t)] \equiv 0$. Suppose that the covariance function $ r(t)$ satisfies the following conditions.

(a) $ r(t) = 1 - \vert t{\vert^\alpha }H(t) + o(\vert t{\vert^\alpha }H(t))$ as $ t \to 0$, where $ 0 < \alpha \leqslant 2$ and $ H$ varies slowly at zero, and

(b) $ r(t) = O(1/\log t)$ as $ t \to \infty $ Then for any nondecreasing positive function $ \phi (t)$ defined on $ [a,\infty )$ with $ \phi (\infty ) = \infty ,P[X(t) > \phi (t)$ i.o. for some sequence $ {t_n} \to \infty ] = 0or1$ according as the integral $ I(\phi ) = \int_a^\infty {g(\phi (t))\phi {{(t)}^{ - 1}}\exp ( - {\phi ^2}(t)/2)dt} $ is finite or infinite, where $ g(x) = 1/_\sigma ^{ \sim - 1}(1/x)$ is a regularly varying function with exponent $ 2/\alpha $ and $ _\sigma ^{ \sim 2}(t) = 2\vert t{\vert^\alpha }H(t)$.

Theorem C. Let $ \{ {X_n}:n \geqslant 1\} $ be a stationary Gaussian sequence with zero mean and unit variance. Suppose that its covariance function satisfies, for some $ \gamma > 0,r(n) = O(1/{n^\gamma })\;as\;n \to \infty $. Let $ \{ \phi (n):n \geqslant 1\} $ be a nondecreasing sequence of positive numbers with $ {\lim _{n \to \infty }}\phi (n) = \infty $; suppose that $ \Sigma (1/\phi (n))\exp ( - {\phi ^2}(n)/2) = \infty $. Then

$\displaystyle \mathop {\lim }\limits_{n \to \infty } \sum\limits_{1 \leq k \leq n} {{I_k}} /\sum\limits_{1 \leq k \leq n} {E[{I_k}] = 1\quad a.s.,} $

where $ {I_k}$ denotes the indicator function of the event $ \{ {X_k} > \phi (k)\} $.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1973-0321170-8
Keywords: Stationary process, Gaussian process, law of iterated logarithm, zero-one law, upcrossings, rate of growth of upcrossings
Article copyright: © Copyright 1973 American Mathematical Society