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Exact tail asymptotics of the supremum of strongly dependent Gaussian processes over a random interval

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Abstract

Let \( \mathcal{T} \) be a positive random variable independent of a real-valued stochastic process \( \left\{ {X(t),t\geqslant 0} \right\} \). In this paper, we investigate the asymptotic behavior of \( \mathrm{P}\left( {{\sup_{{t\in \left[ {0,\mathcal{T}} \right]}}}X(t)>u} \right) \) as u→∞ assuming that X is a strongly dependent stationary Gaussian process and \( \mathcal{T} \) has a regularly varying survival function at infinity with index λ ∈ [0, 1). Under asymptotic restrictions on the correlation function of the process, we show that \( \mathrm{P}\left( {{\sup_{{t\in \left[ {0,\mathcal{T}} \right]}}}X(t)>u} \right)={c^{\lambda }}\mathrm{P}\left( {\mathcal{T}>m(u)} \right)\left( {1+o(1)} \right) \) with some positive finite constant c and function m(·) defined in terms of the local behavior of the correlation function and the standard Gaussian distribution.

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Authors and Affiliations

  1. College of Mathematics, Physics and Information Engineering, Jiaxing University, Jiaxing, 314001, PR China

    Zhongquan Tan

  2. Department of Actuarial Science, Faculty of Business and Economics, University of Lausanne, UNIL-Dorigny, 1015, Lausanne, Switzerland

    Enkelejd Hashorva

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  1. Zhongquan Tan
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  2. Enkelejd Hashorva
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Correspondence to Zhongquan Tan.

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2The author is supported by the Swiss National Science Foundation, grant 200021-1401633/1.

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Tan, Z., Hashorva, E. Exact tail asymptotics of the supremum of strongly dependent Gaussian processes over a random interval. Lith Math J 53, 91–102 (2013). https://doi.org/10.1007/s10986-013-9196-6

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  • DOI: https://doi.org/10.1007/s10986-013-9196-6

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