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 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 $\operatorname {PD}(α,θ)$-bridges. In particular we obtain the best known results for the density of the time spent positive of a Bessel bridge of dimension $2-2α$.
Large deviations related to Dynkin--Lamperti arcsine laws for last visit times of Markov processes
Faculty of Arts and Sciences, Kyoto Institute of Technology（京都工艺纤维大学艺术与科学学院）；Department of Mathematics, Graduate School of Science, The University of Osaka（大阪大学理学研究科数学系）；Department of Systems Innovation, Graduate School of Engineering Science, The University of Osaka（大阪大学工学研究科系统创新系）
Deep factorisation of the stable process III: Radial excursion theory and the point of closest reach
在本文中，我们继续从自相似马尔可夫过程理论的角度来理解稳定过程，这是基于Kyprianou（2016）和Kyprianou等人（2017）最近论文的精神。特别地，我们将注意力转向$d$维各向同性稳定过程的情形，对于$d\geq 2$ .使用一种全新的方法，我们考虑最近到达点的分布。这让我们为这类稳定过程得出了许多其他实质性的新结果。我们从事一个新的径向偏移理论，以前从未使用过，从我们开发的经典Blumenthal-Getoor射线身份首次进入/退出到一个球，参见。Blumenthal et al.（1961），to the setting of $n$ -tuple laws.我们明确确定的平稳分布的稳定过程时，反映在其运行的径向上确界。此外，我们提供了一个表示的Wiener-Hopf分解的MAP的基础上的稳定的过程，通过Lamperti-Kiu变换。
