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

Let { χ k ( 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 { sup t ∈ [ 0 , T ] χ k ( 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 p th-mean theorem for stationary χ -processes.