AI 中文总结
研究独立、离散、整数值且分布具周期性的随机变量序列，推导出\(\max\{X_i,X_i + X_{i + 1},\ldots\}\)分布函数的可计算表示，基于线性递推及相关线性系统确定初始值，还给出例子并纠正文献结果。
AI 中文摘要
设\(X_1,X_2,\ldots,X_N\)（\(N\in\mathbb N\)）为独立、离散、整数值随机变量，各\(X_j\geqslant m_j\)几乎必然成立，且\(m_1+\cdots+m_N<0\)，序列\(X_1,X_2,\ldots\)分布呈周期性。我们推导出\(\max\{X_1,X_1 + X_2,\ldots\}\)等分布函数的可计算表示，公式基于线性递推，初始值由含相关特征方程根及\(X_1,\ldots,X_N\)分布的线性系统确定。给出多个例子并纠正了文献中的一些结果。
英文摘要
Let $X_1,\,X_2,\,\ldots,\,X_N$, $N\in\mathbb N$ be independent, discrete, integer-valued random variables. Assume that $X_j\geqslant m_j$ almost surely for each $j=1,\,2,\,\ldots,\,N$, where $m_1,\,m_2,\,\ldots,\,m_N\in\mathbb{Z}$ satisfy $m_1+\cdots+m_N<0$. Furthermore, suppose that the sequence $X_1,\,X_2,\,\ldots$ is periodic in distribution, i.e. $X_k{\buildrel d \over =} X_{k+N}$ for all $k\in\mathbb N$. We derive computable representations for the distribution functions of $\max\{X_1,\,X_1+X_2,\,\ldots\}$, $\max\{X_2,\,X_2+X_3,\,\ldots\}$, $\ldots$, $\max\{X_N,\,X_N+X_{N+1},\,\ldots\}$. The obtained formulas are based on a linear recurrence whose initial values are determined from a linear system that involves the roots of an associated characteristic equation and the distributions of $X_1,\,X_2,\,\ldots,\,X_N$. Several examples are presented, including a biseasonal-biased Rademacher random walk for which the distribution, generating functions, and all moments admit explicit closed-form expressions. In addition, we identify and correct several inaccuracies in the results reported in \cite{Grigutis2024}.
