Extremes of [formula omitted]-Ornstein–Uhlenbeck processes

Abstract

Two limit theorems are established on the extremes of a family of stationary Markov processes, known as q-Ornstein–Uhlenbeck processes with q∈(−1,1). Both results are crucially based on the weak convergence of the tangent process at the lower boundary of the domain of the process, a positive self-similar Markov process little investigated so far in the literature. The first result is the asymptotic excursion probability established by the double-sum method, with an explicit formula for the Pickands constant in this context. The second result is a Brown–Resnick-type limit theorem on the minimum process of i.i.d. copies of the q-Ornstein–Uhlenbeck process: with appropriate scalings in both time and magnitude, a new semi-min-stable process arises in the limit.