$\mathbf{Z}_+$上的马尔科夫分支随机漫步:使用正交多项式的方法I

IF 0.5 4区 数学 Q4 STATISTICS & PROBABILITY
A. V. Lyulintsev
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引用次数: 0

摘要

概率论及其应用》(Theory of Probability &Its Applications),第 69 卷第 1 期,第 71-87 页,2024 年 5 月。 我们考虑状态空间 $\mathbf{Z}_+=\{0,1,2,\dots\}$ 上的连续时间同质马尔可夫过程。该过程被解释为粒子的运动。粒子只能在相邻点 $\mathbf{Z}_+$ 上运动,也就是说,粒子每运动一次,其坐标就会变化 1。分支源可能位于 $\mathbf{Z}_+$ 的每个点。在发生分支的时刻,新粒子会出现在分支点,然后按照与初始粒子相同的规则相互独立地(以及独立于其他粒子)演化。这种分支马尔可夫过程对应一个雅可比矩阵。通过与该矩阵相对应的正交多项式,我们得到了在时间 $t>0$ 时 $\mathbf{Z}_+$ 的任意固定点上粒子的平均数量公式。我们将所得结果应用于一些具体模型,给出了粒子平均数量的特殊函数精确值,并找到了该数量在大时间内的渐近公式。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Markov Branching Random Walks on $\mathbf{Z}_+$: An Approach Using Orthogonal Polynomials. I
Theory of Probability &Its Applications, Volume 69, Issue 1, Page 71-87, May 2024.
We consider a continuous-time homogeneous Markov process on the state space $\mathbf{Z}_+=\{0,1,2,\dots\}$. The process is interpreted as the motion of a particle. A particle may transit only to neighboring points $\mathbf{Z}_+$, i.e., for each single motion of the particle, its coordinate changes by 1. The process is equipped with a branching mechanism. Branching sources may be located at each point of $\mathbf{Z}_+$. At a moment of branching, new particles appear at the branching point and then evolve independently of each other (and of the other particles) by the same rules as the initial particle. To such a branching Markov process there corresponds a Jacobi matrix. In terms of orthogonal polynomials corresponding to this matrix, we obtain formulas for the mean number of particles at an arbitrary fixed point of $\mathbf{Z}_+$ at time $t>0$. The results obtained are applied to some concrete models, an exact value for the mean number of particles in terms of special functions is given, and an asymptotic formula for this quantity for large time is found.
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来源期刊
Theory of Probability and its Applications
Theory of Probability and its Applications 数学-统计学与概率论
CiteScore
1.00
自引率
16.70%
发文量
54
审稿时长
6 months
期刊介绍: Theory of Probability and Its Applications (TVP) accepts original articles and communications on the theory of probability, general problems of mathematical statistics, and applications of the theory of probability to natural science and technology. Articles of the latter type will be accepted only if the mathematical methods applied are essentially new.
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