LAMPERTI-TYPE LAWS

• 成果类型: Article
• 署名作者: James, Lancelot F.
• 署名单位: Hong Kong University of Science & Technology
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/09-AAP660
• 发表日期: 2010
• 页码: 1303-1340
• 关键词: mittag-leffler functions; random-variables; time; linnik; gamma; REPRESENTATIONS; distributions; asymptotics; PRODUCTS; beta

摘要

This paper explores various distributional aspects of random variables defined as the ratio of two independent positive random variables where one variable has an alpha-stable law, for 0 < alpha < 1, and the other variable has the law defined by polynomially tilting the density of an alpha-stable random variable by a factor theta > -alpha. When theta = 0, these variables equate with the ratio investigated by Lamperti [Trans. Amer. Math. Soc. 88 (1958) 380-387] which, remarkably, was shown to have a simple density. This variable arises in a variety of areas and gains importance from a close connection to the stable laws. This rationale, and connection to the PD(alpha, theta) distribution, motivates the investigations of its generalizations which we refer to as Lamperti-type laws. We identify and exploit links to random variables that commonly appear in a variety of applications. Namely Linnik, generalized Pareto and z-distributions. In each case we obtain new results that are of potential interest. As some highlights, we then use these results to (i) obtain integral representations and other identities for a class of generalized Mittag-Leffler functions, (ii) identify explicitly the Levy density of the semigroup of stable continuous state branching processes (CSBP) and hence corresponding limiting distributions derived in Slack and in Zolotarev [Z. Wahrsch. Verw. Gebiete 9 (1968) 139-145, Teor. Veroyatn. Primen. 2 (1957) 256-266], which are related to the recent work by Berestycki, Berestycki and Schweinsberg, and Bertoin and LeGall [Ann. Inst. H. Poincare Probab. Statist. 44 (2008) 214-238, Illinois J. Math. 50 (2006) 147-181] on beta coalescents. (iii) We obtain explicit results for the occupation time of generalized Bessel bridges and some interesting stochastic equations for PD(alpha, theta)-bridges. In particular we obtain the best known results for the density of the time spent positive of a Bessel bridge of dimension 2 - 2 alpha.