Extremes of $q$-Ornstein-Uhlenbeck processes

The $q$-Ornstein-Uhlenbeck processes, $q\in(-1,1)$, are a family of stationary Markov processes that converge weakly to the standard Ornstein-Uhlenbeck process as $q$ tends to 1. It has been noticed recently that in terms of path properties, however, for each $q$ fixed the $q$-Ornstein-Uhlenbeck process behaves qualitatively different from their Gaussian counterpart in several aspects. Here, two limit theorems on the extremes of $q$-Ornstein-Uhlenbeck processes are established. Both results are based on the weak convergence of the tangent process at the lower boundary, 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. With appropriate scalings in both time and magnitude, a new semi-min-stable process arises in the limit.