Size bias, sampling, the waiting time paradox, and infinite divisibility: when is the increment independent?

Scaling Exponent for Incremental Records

We investigate records in a growing sequence of identical and independently distributed random variables. The record equals the largest value in the sequence, and our focus is on the increment, defined as the difference between two successive records. We investigate sequences in which all increments decrease monotonically, and find that the fraction I_N of sequences that exhibit this property decays algebraically with sequence length N, namely I_N ~ N^{-nu} as N --> infinity. We analyze the case where the random variables are drawn from a uniform distribution with compact support, and obtain t…

Size bias for one and all

Size bias occurs famously in waiting-time paradoxes, undesirably in sampling schemes, and unexpectedly in connection with Stein's method, tightness, analysis of the lognormal distribution, Skorohod embedding, infinite divisibility, and number theory. In this paper we review the basics and survey some of these unexpected connections.

Correlated continuous time random walks and fractional Pearson diffusions

Continuous time random walks have random waiting times between particle jumps. We define the correlated continuous time random walks (CTRWs) that converge to fractional Pearson diffusions (fPDs). The jumps in these CTRWs are obtained from Markov chains through the Bernoulli urn-scheme model and Wright-Fisher model. The jumps are correlated so that the limiting processes are not Lévy but diffusion processes with non-independent increments. The waiting times are selected from the domain of attraction of a stable law

带独立增量的CMJ分支过程与优先连接树中的持久枢纽

Persistent hubs in CMJ branching processes with independent increments and preferential attachment trees

该研究推导了带独立出生等待时间的Crump-Mode-Jagers分支过程谱系树中持久枢纽的存在性、非存在性及唯一性判据，并改进了广义优先连接树的相关结果，方法自包含不依赖该分支过程的已有结论。

Two parallel insurance lines with simultaneous arrivals and risks correlated with inter-arrival times

We investigate an insurance risk model that consists of two reserves which receive income at fixed rates. Claims are being requested at random epochs from each reserve and the interclaim times are generally distributed. The two reserves are coupled in the sense that at a claim arrival epoch, claims are being requested from both reserves and the amounts requested are correlated. In addition, the claim amounts are correlated with the time elapsed since the previous claim arrival. We focus on the probability that this bivariate reserve process survives indefinitely. The infinite- horizon survival…

非马尔可夫增长过程中的领导地位固定

Fixation of leadership in non-Markovian growth processes

本文研究了非马尔可夫增长过程中智能体领导地位的固定问题，证明了领导地位以概率零或一发生并给出充要条件，同时改进了带反馈的“球入盒”模型。