Exact tail asymptotics of the supremum of strongly dependent Gaussian processes over a random interval |
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Authors: | Zhongquan Tan Enkelejd Hashorva |
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Institution: | 1. College of Mathematics, Physics and Information Engineering, Jiaxing University, Jiaxing, 314001, PR China 2. Department of Actuarial Science, Faculty of Business and Economics, University of Lausanne, UNIL-Dorigny, 1015, Lausanne, Switzerland
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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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