The limit law of the maximum of discrete partial-sums distribution II

Abstract

Let $X_1,X_2,\dots ,X_N$, $N\in \mathbb{N}$ be independent, discrete, integer-valued random variables. Assume that $X_j⩾m_j$ almost surely for each $j=1,2,\dots ,N$, where $m_1,m_2,\dots ,m_N\in \mathbb{Z}$ satisfy $m_1+\cdots +m_N<0$. Furthermore, suppose that the sequence $X_1,X_2,\dots$ 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 $\max \{X_1,X_1+X_2,\dots \}$, $\max \{X_2,X_2+X_3,\dots \}$, $\dots$, $\max \{X_N,X_N+X_{N+1},\dots \}$. 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 $X_1,X_2,\dots ,X_N$. 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}.