Multi seasonal discrete time risk model revisited

In this work we set up the distribution function of $\mathcal{M}:={\sup }_{n⩾1}{\sum }_{i=1}^n(Z_i-1)$, where the random walk ${\sum }_{i=1}^n{Z}_i,n\in \mathbb{N},$ is generated by $N$ periodically occurring distributions and the integer-valued and non-negative random variables $Z_1,Z_2,\dots$ are independent. The considered random walk generates so-called multi seasonal discrete time risk model, and a known distribution of random variable $\mathcal{M}$ enables to calculate ultimate time ruin or survival probability. Verifying obtained theoretical statements we demonstrate several computational examples for survival probability $\mathbb{P}(\mathcal{M}<u)$ when $N=2,3$ or $10$.