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December, 1972 An Exponential Probability Bound for the Energy of a Type of Gaussian Process
L. H. Koopmans, C. Qualls
Ann. Math. Statist. 43(6): 1953-1960 (December, 1972). DOI: 10.1214/aoms/1177690866

Abstract

Real-valued stochastic processes of the form $x(t) = \int A(t, \lambda)Z(d\lambda)$ are considered, where $Z(\lambda)$ is a zero mean Gaussian process with independent increments and $\int\int |A(t, \lambda)|^2F(d\lambda) dt < \infty$, where $F(d\lambda) = E|Z(d\lambda)|^2$. It is shown that the energy of $x(t), \int x^2(t) dt$, is a well-defined random variable and an exponential bound for $P(\int x^2(t) dt - E\int x^2(t) dt \geqq \varepsilon)$ is derived. This bound is used to obtain an exponential bound for crossing probabilities $P(|y(t)| > a$ for some $t)$ where $y(t) = \int h(t - \tau)x(\tau) d\tau, \int h^2(t) dt < \infty$.

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L. H. Koopmans. C. Qualls. "An Exponential Probability Bound for the Energy of a Type of Gaussian Process." Ann. Math. Statist. 43 (6) 1953 - 1960, December, 1972. https://doi.org/10.1214/aoms/1177690866

Information

Published: December, 1972
First available in Project Euclid: 27 April 2007

zbMATH: 0248.60032
MathSciNet: MR362467
Digital Object Identifier: 10.1214/aoms/1177690866

Rights: Copyright © 1972 Institute of Mathematical Statistics

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Vol.43 • No. 6 • December, 1972
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