Uniform convergence of exact large deviations for renewal reward processes

• 成果类型: Article
• 署名作者: Chi, Zhiyi
• 署名单位: University of Connecticut
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/105051607000000023
• 发表日期: 2007
• 页码: 1019-1048
• 关键词: boundary crossing probabilities; markov additive processes; LIMIT-THEOREMS; asymptotic expansions

摘要

Let (X-n, Y-n) be i.i.d. random vectors. Let W(x) be the partial sum of Yn just before that of X-n exceeds x > 0. Motivated by stochastic models for neural activity, uniform convergence of the form sup(c is an element of I) vertical bar a(c, x) Pr{W(x) >= cx} - 1 vertical bar = o(1), x -> infinity, is established for probabilities of large deviations, with a (c, x) a deterministic function and I an open interval. To obtain this uniform exact large deviations principle (LDP), we first establish the exponentially fast uniform convergence of a family of renewal measures and then apply it to appropriately tilted distributions Of X-n and the moment generating function of W(x). The uniform exact LDP is obtained for cases where X-n has a subcomponent with a smooth density and Y-n is not a linear transform of X-n. An extension is also made to the partial sum at the first exceedance time.