On perpetuities related to the size-biased distributions

A log-type moment result for perpetuities and its application to martingales in supercritical branching random walks

Infinite sums of i.i.d. random variables discounted by a multiplicative random walk are called perpetuities and have been studied by many authors. The present paper provides a log-type moment result for such random variables under minimal conditions which is then utilized for the study of related moments of a.s. limits of certain martingales associated with the supercritical branching random walk. The connection, first observed by the second author in [Iksanov, A.M. (2004). Elementary fixed points of the BRW smoothing transforms with infinite number of summands. Stoch. Proc. Appl. 114, 27-50.]…

New explicit examples of fixed points of Poisson shot noise transforms

We show that gamma distributions, generalized positive Linnik distributions, S2 distributions are fixed points of Poisson shot noise transforms. The corresponding response functions are identified via their inverse functions except for some special cases when those can be obtained explicitly. As a by-product, it is proven that log-convexity of the response function is not necessary for selfdecomposability of non-negative Poisson shot noise distribution. Some attention is given to perpetuities of a rather special type which are closely related to our model. In particular, we study the problem o…

一类广义的规模有偏分布

A General size-biased distribution

本研究借助Ferrar的成果推广了与黎曼ξ函数相关的规模有偏分布，并给出了这类更广义分布的若干分析结论与性质，丰富了相关概率分布的研究内容。

Size-biased permutation of a finite sequence with independent and identically distributed terms

This paper focuses on the size-biased permutation of $n$ independent and identically distributed (i.i.d.) positive random variables. This is a finite dimensional analogue of the size-biased permutation of ranked jumps of a subordinator studied in Perman-Pitman-Yor (PPY) [Probab. Theory Related Fields 92 (1992) 21-39], as well as a special form of induced order statistics [Bull. Inst. Internat. Statist. 45 (1973) 295-300; Ann. Statist. 2 (1974) 1034-1039]. This intersection grants us different tools for deriving distributional properties. Their comparisons lead to new results, as well as simple…

基于度数偏好的随机递归树切割过程

A degree-biased cutting process for random recursive trees

本文提出了一种基于度数偏好的随机递归树切割过程，推导了切割次数的递归公式，并证明其收敛性与稳定分布的关系。

尺寸偏置扩散极限与包含过程

Size-biased diffusion limits and the inclusion process

本文研究完全图上扩散率消失的包含过程，利用尺寸偏置经验测度建立其向测度值马尔可夫过程的收敛，统一覆盖各标度区域并推广Poisson-Dirichlet扩散至无穷突变率。