Asymptotic expansions of distributions in two-boundary problems for stochastic processes with independent increments

摘要

We consider a homogeneous stochastic process #xi#(t), t >= 0, #xi#(0) = 0, with independent increments, E#xi#(1) not= 0. Under Cramer's type conditions on #xi#(1), we obtain full asymptotic expansions for joinst distributions of random variables T and #xi#(T), where T = inf{t > 0: #xi#(t) #epsilon# [-a, b)}, a > 0, b > 0. The results are obtained for the case Ct~(1/2) <= a + b = o(t) -> infinity as t -> infinity.