Lindley递推式的一种乘法形式
A Multiplicative Version of the Lindley Recursion
摘要：
本文提出了一种分析的随机递归$W_{i+1} = [V_iW_i+Y_i]^+$，可以解释为一个自回归过程的顺序1，反映在0。我们首先讨论模型的稳定性条件。写入$Y_i=B_i-A_i$，用于非负i.i.d.的独立序列\随机变量$\{A_i\}_{i\in N_0}$和$\{B_i\}_{i\in N_0}$，并假设$\{V_i\}_{i\in N_0}$是i.i.d.序列以及（独立于$\{A_i\}_{i\in N_0}$和$\{B_i\}_{i\in N_0}$），我们考虑三种特殊情况：（i）$V_i$仅获得负值，并且$B_i$具有有理LST，（ii）$V_i$以一定概率$p\in (0,1)$等于正值$a$，否则为负，并且$A_i$和$B_i$都具有有理LST，（iii）$V_i$均匀分布在$[0,1]$上，并且$A_i$是指数分布的。在所有这三种情况下，我们得到瞬态和稳态的结果，其中瞬态的结果是在一个几何分布的时代的变换。
英文摘要：
This paper presents an analysis of the stochastic recursion $W_{i+1} = [V_iW_i+Y_i]^+$ that can be interpreted as an autoregressive process of order 1, reflected at 0. We start our exposition by a discussion of the model's stability condition. Writing $Y_i=B_i-A_i$, for independent sequences of non-negative i.i.d.\ random variables $\{A_i\}_{i\in N_0}$ and $\{B_i\}_{i\in N_0}$, and assuming $\{V_i\}_{i\in N_0}$ is an i.i.d. sequence as well (independent of $\{A_i\}_{i\in N_0}$ and $\{B_i\}_{i\in N_0}$), we then consider three special cases: (i) $V_i$ attains negative values only and $B_i$ has a rational LST, (ii) $V_i$ equals a positive value $a$ with certain probability $p\in (0,1)$ and is negative otherwise, and both $A_i$ and $B_i$ have a rational LST, (iii) $V_i$ is uniformly distributed on $[0,1]$, and $A_i$ is exponentially distributed. In all three cases we derive transient and stationary results, where the transient results are in terms of the transform at a geometrically distributed epoch.
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