This paper concerns a modified version of the Lindley recursion, where the recursion equation is given by \(W_{i+1} = [V_{i} W_{i} + Y_{i}]^{+}\), with \(\{V_i\}_{i=0}^{\infty }\) and \(\{Y_i\}_{i=0}^{\infty }\) being two independent sequences of i.i.d. random variables. Additionally, we assume that the \(V_i\) take values in \((-\infty , 1]\) and the \(Y_i\) have a rational Laplace–Stieltjes transform. Under these assumptions, we investigate the transient and steady-state behaviors of the process \(\{W_i\}_{i=0}^{\infty }\) by deriving an expression for the generating function of the Laplace–Stieltjes transform of the \(W_i\).

本文涉及 Lindley 递归的修改版本，其中递归方程由\(W_{i+1} = [V_{i} W_{i} + Y_{i}]^{+}\)给出，其中\(\{V_i\}_{i=0}^{\infty }\)和\(\{Y_i\}_{i=0}^{\infty }\)是两个独立的 iid 随机变量序列。此外，我们假设\(V_i\)取\((-\infty , 1]\)中的值，并且\(Y_i\)具有有理Laplace-Stieltjes变换。在这些假设下，我们研究瞬态和稳态- 通过推导\(W_i\) 的拉普拉斯-斯蒂尔杰斯变换的生成函数表达式来状态过程\( \{W_i\ }_{i=0}^{\infty }\) 的行为。