2016/09/01 by Yizao Wang, Wang, Yizao
Mathematics · #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.1609.00338
openalex publication_date 2016/09/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The q-Ornstein-Uhlenbeck processes, q∈(-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.