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

Published in at http://dx.doi.org/10.1214/09-AAP660 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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• On the fractional counterpart of the higher-order equations
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• Further examples of GGC and HCM densities
• Random weighted averages, partition structures and generalized arcsine laws
• Dirichlet mean identities and laws of a class of subordinators
• Bessel processes and hyperbolic Brownian motions stopped at different random times
• Generalized Mittag Leffler distributions arising as limits in preferential attachment models
• Quantile clocks
• Further examples with moments of Gamma type
• Phase transitions of composition schemes: Mittag-Leffler and mixed Poisson distributions
• Coag-Frag duality for a class of stable Poisson-Kingman mixtures
• Stick-breaking Pitman-Yor processes given the species sampling size
• Wright functions governed by fractional directional derivatives and fractional advection diffusion equations
• Fractional Poisson process with random drift
• Some properties of stationary continuous state branching processes
• Extremes and gaps in sampling from a GEM random discrete distribution
• Occupation time of a renewal process coupled to a discrete Markov chain
• Gibbs Partitions, Riemann-Liouville Fractional Operators, Mittag-Leffler Functions, and Fragmentations Derived From Stable Subordinators