Multiseasonal discrete-time risk model revisited

In this work, we set up the distribution function of M:=supn≥1∑i=1nXi-1,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{M}:={\mathrm{sup}}_{n\ge 1}{\sum }_{i=1}^{n}\left({X}_{i}-1\right),$$\end{document} where the random walk ∑i=1nXi,n∈N,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\sum }_{i=1}^{n}{X}_{i},n\in {\mathbb{N}},$$\end{document} is generated by N periodically occurring distributions, and the integer-valued and nonnegative random variablesX1,X2, . . . are independent. The considered random walk generates a so-called multiseasonal discrete-time risk model, and a known distribution of random variable M enables us to calculate the ultimate time ruin or survival probability. Verifying obtained theoretical statements, we demonstrate several computational examples for survival probability P(M < u) when N = 2, 3, or 10.