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Extremes of \(q\)-Ornstein-Uhlenbeck processes. (English) Zbl 1405.60072
Summary: Two limit theorems are established on the extremes of a family of stationary Markov processes, known as \(q\)-Ornstein-Uhlenbeck processes with \(q \in(- 1, 1)\). 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.

MSC:
60G70 Extreme value theory; extremal stochastic processes
60J25 Continuous-time Markov processes on general state spaces
Software:
GAUSSIAN
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