Multiseasonal discrete-time risk model revisited

In this work, we set up the distribution function of \(\mathcal{M}:={\mathrm{sup}}_{n\ge 1}{\sum }_{i=1}^{n}\left({X}_{i}-1\right),\) where the random walk \({\sum }_{i=1}^{n}{X}_{i},n\in {\mathbb{N}},\) 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.