摘要翻译：
设$x_1,X_2,...$是一个随机变量序列，满足分布递归$x_1=0$和$x_n=X_{n-I_n}+1$对于$n=2,3,...$,其中$i_n$是一个随机变量，其值在$\1,...，n-1\}$中，它与$X_2,...，X_{n-1}$无关。随机变量$x_n$可以解释为状态空间${\mathbb N}:={1,2,...}$,吸收状态为1的适当死亡马尔可夫链的吸收时间，条件是该链开始于初始状态$N$。本文着重讨论了当$N$趋于无穷大时，$i_n$的分布满足${\mathbb P}\{i_n=k\}=p_k/(P_1+...+P_{n-1})$的特殊而重要的假设，即对于某些给定的概率分布$p_k={\mathbb P}\{\xi=k\}$,$k\在{\mathbb N}$中。根据$\xi$分布的尾部性质，对$x_n$和相应的极限分布进行了几种标度，其中包括稳定分布和从属子指数积分分布。本文所用的方法主要是概率的。关键工具是一种耦合技术，它将$x_n$的分布与随机游动联系起来，例如，它解释了Mittag-Leffler分布在本文中的出现。所得结果用于描述某些β聚结过程的碰撞数的渐近性。
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英文标题：
《On a random recursion related to absorption times of death Markov chains》
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作者：
Alex Iksanov and Martin M\"ohle
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最新提交年份：
2007
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分类信息：

一级分类：Mathematics        数学
二级分类：Probability        概率
分类描述：Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory
概率论与随机过程的理论与应用：例如中心极限定理，大偏差，随机微分方程，统计力学模型，排队论
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一级分类：Mathematics        数学
二级分类：Statistics Theory        统计理论
分类描述：Applied, computational and theoretical statistics: e.g. statistical inference, regression, time series, multivariate analysis, data analysis, Markov chain Monte Carlo, design of experiments, case studies
应用统计、计算统计和理论统计：例如统计推断、回归、时间序列、多元分析、数据分析、马尔可夫链蒙特卡罗、实验设计、案例研究
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一级分类：Statistics        统计学
二级分类：Statistics Theory        统计理论
分类描述：stat.TH is an alias for math.ST. Asymptotics, Bayesian Inference, Decision Theory, Estimation, Foundations, Inference, Testing.
Stat.Th是Math.St的别名。渐近，贝叶斯推论，决策理论，估计，基础，推论，检验。
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英文摘要：
  Let $X_1,X_2,...$ be a sequence of random variables satisfying the distributional recursion $X_1=0$ and $X_n= X_{n-I_n}+1$ for $n=2,3,...$, where $I_n$ is a random variable with values in $\{1,...,n-1\}$ which is independent of $X_2,...,X_{n-1}$. The random variable $X_n$ can be interpreted as the absorption time of a suitable death Markov chain with state space ${\mathbb N}:=\{1,2,...\}$ and absorbing state 1, conditioned that the chain starts in the initial state $n$.   This paper focuses on the asymptotics of $X_n$ as $n$ tends to infinity under the particular but important assumption that the distribution of $I_n$ satisfies ${\mathbb P}\{I_n=k\}=p_k/(p_1+...+p_{n-1})$ for some given probability distribution $p_k={\mathbb P}\{\xi=k\}$, $k\in{\mathbb N}$. Depending on the tail behaviour of the distribution of $\xi$, several scalings for $X_n$ and corresponding limiting distributions come into play, among them stable distributions and distributions of exponential integrals of subordinators. The methods used in this paper are mainly probabilistic. The key tool is a coupling technique which relates the distribution of $X_n$ to a random walk, which explains, for example, the appearance of the Mittag-Leffler distribution in this context. The results are applied to describe the asymptotics of the number of collisions for certain beta-coalescent processes.
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PDF链接：
https://arxiv.org/pdf/710.5826