Exact asymptotics and limit theorems for supremum of stationary χχ-processes over a random interval

Let {χk(t),t≥0}{χk(t),t≥0} be a stationary χχ-process with kk degrees of freedom being independent of some non-negative random variable TT. In this paper we derive the exact asymptotics of P{supt∈[0,T]χk(t)>u}P{supt∈[0,T]χk(t)>u} as u→∞u→∞ when TT has a regularly varying tail with index λ∈[0,1)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 ppth-mean theorem for stationary χχ-processes.