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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 $α$-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 $\operatorname {PD}(α,θ)$ 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 den