Lamperti-type laws

Abstract

This paper explores various distributional aspects of random variables defined as the ratio of two independent positive random variables where one variable has an α-stable law, for 0 < α < 1, and the other variable has the law defined by polynomially tilting the density of an α-stable random variable by a factor θ > −α. When θ = 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 (α, θ) 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 Lévy 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. Poincaré 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 (α, θ)-bridges. In particular we obtain the best known results for the density of the time spent positive of a Bessel bridge of dimension 2 − 2α.