Persistence of AR(1)-sequences.
1885-е заседание.
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Vitali Wachtel (Institut für Mathematik Universität Augsburg)
Persistence of AR(1)-sequences.
Let \xi_k be independent, identically distributed random variables and let a\in(0,1) be a fixed constant. An AR(1)-sequence is defined by X_n=aX_{n-1}+\xi_n, n>= 1, where the starting point X_0 of this process may be either deterministic or distributed according to any probabilistic measure \nu. We are interested in the tail behaviour of the stopping time T_0:=\min{k >= 1 : X_k<= 0}. We find minimal moment conditions on the innovations, under which one has P_x(T_0>n)\sim V(x)e^{-\lambda n}, as n\to\infty, where \lambda is a positive number.
Источник: Объединенный семинар «Вероятность и математическая статистика»