THE ASYMPTOTIC BEHAVIOR OF DENSITIES RELATED TO THE SUPREMUM OF A STABLE PROCESS

• 成果类型: Article
• 署名作者: Doney, R. A.; Savov, M. S.
• 署名单位: University of Manchester; Sorbonne Universite
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/09-AOP479
• 发表日期: 2010
• 页码: 316-326
• 关键词: Levy processes

摘要

If X is a stable process of index alpha is an element of (0, 2) whose Levy measure has density cx(-alpha-1) on (0, infinity), and S(1) = sup(0<= 1) X(t), it is known that P(S(1) > x) similar to A alpha-1 x-alpha as x -> infinity and P(S(1) <= x) similar to B alpha(-1) rho(-1) x(alpha rho) as x down arrow 0. [Here rho = (X(1) > 0) and A and B are known constants.] It is also known that S(1) has a continuous density, m say. The main point of this note is to show that m(x) similar to Ax(-(alpha+1)) as x -> infinity and m(x) similar to Bx(alpha rho-1) as x down arrow 0. Similar results are obtained for related densities.