Lamperti-type laws

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 $\operatorname {PD}(\alpha,\theta)$ 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 L\'{e}vy 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. Poincar\'{e} 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 $\operatorname {PD}(\alpha,\theta)$-bridges. In particular we obtain the best known results for the density of the time spent positive of a Bessel bridge of dimension $2-2\alpha$.