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New bounds for the first passage, wave-length and amplitude densities

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  • Rychlik, Igor

Abstract

Durbin has presented a compact formula for the first passage density of a Gaussian process, which is locally like Brownian motion, to a smooth barrier. In previous work, we have extended the formula to the case of processes which are smooth functions of a continuously differentiable Gaussian vector process. In the present paper we extend the results to more general kinds of first passage time problems, so called marked crossings, and use it to construct upper and lower bounds for the first passage, wave-length densities and for the transition distribution from a maximum to the following minimum. Numerical examples illustrate the results.

Suggested Citation

  • Rychlik, Igor, 1990. "New bounds for the first passage, wave-length and amplitude densities," Stochastic Processes and their Applications, Elsevier, vol. 34(2), pages 313-339, April.
    • Handle: RePEc:eee:spapps:v:34:y:1990:i:2:p:313-339
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    Cited by:

    1. Azaïs, Jean-Marc & Wschebor, Mario, 2008. "A general expression for the distribution of the maximum of a Gaussian field and the approximation of the tail," Stochastic Processes and their Applications, Elsevier, vol. 118(7), pages 1190-1218, July.
    2. Azaïs, Jean-Marc & Pham, Viet-Hung, 2016. "Asymptotic formula for the tail of the maximum of smooth stationary Gaussian fields on non locally convex sets," Stochastic Processes and their Applications, Elsevier, vol. 126(5), pages 1385-1411.

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