Large deviations probabilities for random walks in the absence of finite expectations of jumps

• 成果类型: Article
• 署名作者: Borovkov, AA
• 署名单位: Russian Academy of Sciences; Sobolev Institute of Mathematics
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-002-0243-1
• 发表日期: 2003
• 页码: 421-446
• 关键词: sums

摘要

Let X-1, X-2, . . . be independent identically distributed random variables with regularly varying distribution tails: P(X-1 > t) = V(t) equivalent to t(-beta) L (t), P(X-1 < -t) = W(t) equivalent to t(-alpha) L-W(t), where alpha less than or equal to min(1, beta), and L and L-W are slowly varying functions as t --> infinity. Set S-n = X-1 + (. . .) + X-n, (S) over bar (n) = max(0less than or equal tokless than or equal ton) S-k. We find the asymptotic behavior of P(S-n > x) --> 0 and P(S (S) over bar (n) > x) --> 0 As x --> infinity, give a criterion for (S) over bar (infinity) < infinity a.s. and, under broad conditions, prove that P ((S) over bar (infinity) > x) similar to cV (x)/W (x). In case when distribution tails of X-j admit regularly varying majorants or minorants we find sharp estimates for the mentioned above probabilities under study. We also establish a joint distributional representation for the global maximum. and the time eta when it was attained in the form of a compound Poisson random vector.