Distribution of shifted discrete random walk generated by distinct random variables and applications in ruin theory

In this paper, we set up the distribution function $$\varphi(u)=\mathbb{P}\left(\sup_{n\geqslant 1}\sum_{i=1}^{n}\left(X_i-\kappa\right)<u\right),$$ and the generating function of $\varphi(u+1)$, where $u\in\mathbb{N}_0$, $\kappa\in\mathbb{N}$, the random walk $\left\{\sum_{i=1}^{n}X_i, n\in\mathbb{N}\right\},$ consists of $N\in\mathbb{N}$ periodically occurring distributions, and the integer-valued and non-negative random variables $X_1,\,X_2,\,\ldots$ are independent. This research generalizes two recent works where $\{\kappa=1,\,N\in\mathbb{N}\}$ and $\{\kappa\in\mathbb{N},\,N=1\}$ were considered respectively. The provided sequence of sums $\left\{\sum_{i=1}^{n}\left(X_i-\kappa\right),\,n\in\mathbb{N}\right\}$ generates so-called multi-seasonal discrete-time risk model with arbitrary natural premium and its known distribution enables to calculate the ultimate time ruin probability $1-\varphi(u)$ or survival probability $\varphi(u)$. Verifying obtained theoretical statements we demonstrate several computational examples for survival probability $\varphi(u)$ and its generating function when $\{\kappa=2,\,N=2\}$, $\{\kappa=3,\,N=2\}$, $\{\kappa=5,\,N=10\}$ and $X_i$ admits Poisson and some other distributions. We also conjecture the non-singularity of certain matrices.