Extremes and limit theorems for difference of chi-type processes

Albin, Patrik; Hashorva, Enkelejd; Ji, Lanpeng; Ling, Chengxiu

ESAIM: Probability and Statistics, Tome 20 (2016), pp. 349-366.

Let $\{{\zeta }_{m,k}^{\left(\kappa \right)}\left(t\right),t\ge 0\},\kappa >0$ be random processes defined as the differences of two independent stationary chi-type processes with $m$ and $k$ degrees of freedom. In this paper we derive the asymptotics of $\mathbb{P}\{{sup}_{t\in [0,T]}{\zeta }_{m,k}^{\left(\kappa \right)}\left(t\right)>u\},u\to \infty$ under some assumptions on the covariance structures of the underlying Gaussian processes. Further, we establish a Berman sojourn limit theorem and a Gumbel limit result.

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