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Extreme sojourns of a Gaussian process with a point of maximum variance
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  • Published: March 1987

Extreme sojourns of a Gaussian process with a point of maximum variance

  • Simeon M. Berman1 

Probability Theory and Related Fields volume 74, pages 113–124 (1987)Cite this article

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  • 7 Citations

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Summary

LetX(t), 0≦t≦1, be a Gaussian process with mean 0 and continuous covariance function. Put σ2(t)=EX 2(t), and suppose that there is a point τ in [0, 1] such that σ2(t) has unique maximum att=τ. PutL u =mes(s: 0≦s≦1,X(s)>u). Under suitable conditions there exists a functionv=v(u) and a distribution functionG(x), x>0, such thatv(u)→∞ foru→∞, and

$$\mathop {\lim }\limits_{u \to \infty } \frac{{\int\limits_0^x {ydP(\upsilon L_u \leqq y)} }}{{\upsilon EL_u }} = G(x).$$

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References

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Authors and Affiliations

  1. Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, 10012, New York, NY, USA

    Simeon M. Berman

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  1. Simeon M. Berman
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Additional information

This paper represents results obtained at the Courant Institute of Mathematical Sciences, New York University, under the sponsorship of the National Science Foundation, Grants MCS 82. 01119 and DMS 85 01512

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Cite this article

Berman, S.M. Extreme sojourns of a Gaussian process with a point of maximum variance. Probab. Th. Rel. Fields 74, 113–124 (1987). https://doi.org/10.1007/BF01845642

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  • Received: 28 January 1985

  • Revised: 12 January 1986

  • Issue Date: March 1987

  • DOI: https://doi.org/10.1007/BF01845642

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Keywords

  • Covariance
  • Stochastic Process
  • Probability Theory
  • Mathematical Biology
  • Covariance Function
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