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August, 1972 Maximum and High Level Excursion of a Gaussian Process with Stationary Increments
Simeon M. Berman
Ann. Math. Statist. 43(4): 1247-1266 (August, 1972). DOI: 10.1214/aoms/1177692476


Let $X(t), t \geqq 0$, be a centered separable Gaussian process with stationary increments. Put $\sigma^2(t) = EX^2(t)$, and suppose that $\sigma(0) = 0$. Let $Y(t)$ be the normalized process $X(t)/\sigma(t)$, and $B$ an arbitrary bounded closed subinterval of the positive real axis. Under general conditions on $\sigma$ we find (1) an explicit asymptotic formula for $P\{\max_B Y > u\}$ for $u \rightarrow \infty$ in terms of $\sigma$ and various functions derived from it, and (2) the limiting conditional distribution of the time spent above the level $u$ (for $u \rightarrow \infty$) given that the time spent is positive. This limiting distribution is a scale mixture of the corresponding distribution previously obtained under comparable conditions in the case of the stationary Gaussian process.


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Simeon M. Berman. "Maximum and High Level Excursion of a Gaussian Process with Stationary Increments." Ann. Math. Statist. 43 (4) 1247 - 1266, August, 1972.


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

zbMATH: 0241.60030
MathSciNet: MR314103
Digital Object Identifier: 10.1214/aoms/1177692476

Rights: Copyright © 1972 Institute of Mathematical Statistics

Vol.43 • No. 4 • August, 1972
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