Extrema of multi-dimensional Gaussian processes over random intervals

This paper studies the joint tail asymptotics of extrema of the multi-dimensional Gaussian process over random intervals defined as

P(u) := P{∩ni=1(supt∈[0,Ti](Xi(t)+cit) > aiu)} , u→∞ , where Xi(t) , t ≥ 0 , i=1,2,…,n , are independent centered Gaussian processes with stationary increments, T=(T1,…,Tn) is a regularly varying random vector with positive components, which is independent of the Gaussian processes, and ci ∈ R , ai > 0 , i = 1,2,…,n . Our result shows that the structure of the asymptotics of P(u) is determined by the signs of the drifts ci . We also discuss a relevant multi-dimensional regenerative model and derive the corresponding ruin probability.