二维随机过程的常返性和随机游动离开象限的时间
Recurrence of 2-dimensional queueing processes, and random walk exit times from the quadrant
摘要：
设$X = (X_1, X_2)$为二维随机变量，$X(n), n \in \mathbb{N}$为i.i.d. $X$的副本。相关的随机游动是$S(n)= X(1) + \cdots +X(n)$。第一象限中的相应的床反射行走$W(n), n \in \mathbb{N}$由$W(0) = x \in \mathbb{R}_+^2$和$W(n) = \max \{ 0, W(n-1) - X(n) \}$给出，其中最大值是按坐标方式取的。这通常被称为Lindley过程，并对两个服务器队列中的等待时间进行建模。我们描述这个过程的复发，假设合适的，而温和的时刻条件$X$。结果表明，这与随机游动$x + S(n)$从象限的退出时间的尾渐近性直接相关，因此本文的主要部分致力于分析退出时间与漂移向量的关系，即，对$X$的期望。
英文摘要：
Let $X = (X_1, X_2)$ be a 2-dimensional random variable and $X(n), n \in \mathbb{N}$ a sequence of i.i.d. copies of $X$. The associated random walk is $S(n)= X(1) + \cdots +X(n)$. The corresponding absorbed-reflected walk $W(n), n \in \mathbb{N}$ in the first quadrant is given by $W(0) = x \in \mathbb{R}_+^2$ and $W(n) = \max \{ 0, W(n-1) - X(n) \}$, where the maximum is taken coordinate-wise. This is often called the Lindley process and models the waiting times in a two-server queue. We characterize recurrence of this process, assuming suitable, rather mild moment conditions on $X$. It turns out that this is directly related with the tail asymptotics of the exit time of the random walk $x + S(n)$ from the quadrant, so that the main part of this paper is devoted to an analysis of that exit time in relation with the drift vector, i.e., the expectation of $X$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
