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An asymptotic formula for the distribution of the maximum of a Gaussian process with stationary increments

Published online by Cambridge University Press:  14 July 2016

Simeon M. Berman
Simeon M. Berman*
Affiliation:
Courant Institute of Mathematical Sciences
*
∗Postal addres: Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012, USA.
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Abstract

Let X(t), t≧0, be a Gaussian process with mean 0 and stationary increments. If the incremental variance function σ2(t) is convex and σ2(t) = o(t) for t → 0, then P(max[o,t]X(s) > u) ~ P(X(t) > u) for u → ∞ and each t > 0.

Keywords

SAMPLE FUNCTION MAXIMUM
Type
Short Communications
Information
Journal of Applied Probability , Volume 22 , Issue 2 , June 1985 , pp. 454 - 460
Copyright
Copyright © Applied Probability Trust 1985 

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Footnotes

This paper represents results obtained at the Courant Institute of Mathematical Sciences, New York University, under the sponsorship of the National Science Foundation, Grant MCS 82–01119.

References

[1] Berman, S. M. (1982) Sojourns and extremes of stationary processes. Ann. Prob. 10, 1–46.Google Scholar
[2] Slepian, D. (1962) The one-sided barrier problem for Gaussian noise. Bell System Tech. J. 41, 463–501.Google Scholar