Abstract
In this paper, we find the limit set of a sequence (2 log n)−1/2 X n (t), n≧3) of Gaussian processes in C [0,1], where the processes X n (t) are defined on the same probability space and have the same distribution. Our result generalizes the theorems of Oodaira and Strassen, and we also apply it to obtain limit theorems for stationary Gaussian processes, moving averages of the type \(\int\limits_0^t {f\left( {t - s} \right)dW\left( s \right)} \), where W(s) is the standard Wiener process, and other Gaussian processes. Using certain properties of the unit ball of the reproducing kernel Hubert space of X n (t), we derive the usual law of the iterated logarithm for Gaussian processes. The case of multidimensional time is also considered.
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Research supported by the Office of Naval Research under Contract Number N00014-67-A-0108-0018 at Columbia University.
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Lai, T.L. Reproducing kernel Hilbert spaces and the law of the iterated logarithm for Gaussian processes. Z. Wahrscheinlichkeitstheorie verw Gebiete 29, 7–19 (1974). https://doi.org/10.1007/BF00533181
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DOI: https://doi.org/10.1007/BF00533181