LOGARITHMIC TAILS OF SUMS OF PRODUCTS OF POSITIVE RANDOM VARIABLES BOUNDED BY ONE

• 成果类型: Article
• 署名作者: Kolodziejek, Bartosz
• 署名单位: Warsaw University of Technology
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/16-AAP1228
• 发表日期: 2017
• 页码: 1171-1189
• 关键词: RANDOM DIFFERENCE-EQUATIONS; exponential functionals; Levy processes; thin tails; perpetuities; REPRESENTATION; asymptotics

摘要

In this paper, we show under weak assumptions that for R = (d) 1 + M-1 M1M2 + . . ., where P(M is an element of [0, 1]) = 1 and M-i are independent copies of M, we have lnP(R > x) similar to CxlnP(M > 1 - 1/x) as x -> infinity. The constant C is given explicitly and its value depends on the rate of convergence of 1nP(M > 1 - 1/x). Random variable R satisfies the stochastic equation R 1 + MR with M and R independent, thus this result fits into the study of tails of iterated random equations, or more specifically, perpetuities.