Local probabilities for random walks conditioned to stay positive

• 成果类型: Article
• 署名作者: Vatutin, Vladimir A.; Wachtel, Vitali
• 署名单位: Russian Academy of Sciences; Steklov Mathematical Institute of the Russian Academy of Sciences; Technical University of Munich
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-007-0124-8
• 发表日期: 2009
• 页码: 177-217
• 关键词: limit-theorem; fluctuation; height

摘要

Let S-0 = 0, {S-n, n >= 1} be a random walk generated by a sequence of i.i.d. random variables X-1, X-2,... and let tau(-) = min{n >= 1 : S-n <= 0} and tau(+) = min{n >= 1 : S-n > 0}. Assuming that the distribution of X-1 belongs to the domain of attraction of an alpha-stable law we study the asymptotic behavior, as n -> infinity, of the local probabilities P(tau(+/-) = n) and prove the Gnedenko and Stone type conditional local limit theorems for the probabilities P(S-n is an element of [x, x + Delta)vertical bar tau(-) > n) with fixed Delta and x = x(n) is an element of (0, infinity).