Exact asymptotics and limit theorems for supremum of stationary [formula omitted]-processes over a random interval

Abstract

Let {χk(t),t≥0} be a stationary χ-process with k degrees of freedom being independent of some non-negative random variable T. In this paper we derive the exact asymptotics of P{supt∈[0,T]χk(t)>u} as u→∞ when T has a regularly varying tail with index λ∈[0,1). Three other novel results of this contribution are the mixed Gumbel limit law of the normalised maximum over an increasing random interval, the Piterbarg inequality and the Seleznjev pth-mean theorem for stationary χ-processes.