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Spectral conditions for sojourn and extreme value limit theorems for Gaussian processes

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  • Berman, Simeon M.

Abstract

Let X(t), t[greater-or-equal, slanted]0, be a stationary Gaussian process, and define the sojourn time Lu(t)=mes{s:0 [less-than-or-equals, slant]s[less-than-or-equals, slant]t,X(s)>u} and the maximum Z(t)=max(X(s): 0 [less-than-or-equals, slant]0[less-than-or-equals, slant]s[less-than-or-equals, slant]t). Limit theorems for the distributions of Lu(t) and Z(t), for t, u --> [infinity], are obtained under specified conditions on the spectral density of the process. The results supplement earlier theorems obtained under suitable conditions on the covariance function.

Suggested Citation

  • Berman, Simeon M., 1991. "Spectral conditions for sojourn and extreme value limit theorems for Gaussian processes," Stochastic Processes and their Applications, Elsevier, vol. 39(2), pages 201-220, December.
    • Handle: RePEc:eee:spapps:v:39:y:1991:i:2:p:201-220
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    Cited by:

    1. Seuret, Stéphane & Yang, Xiaochuan, 2019. "On sojourn of Brownian motion inside moving boundaries," Stochastic Processes and their Applications, Elsevier, vol. 129(3), pages 978-994.

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