On the exact survival probability by setting discrete random variables in E. Sparre Andersen's model

In this work, we propose a simplification of the Pollaczek-Khinchine formula for the ultimate time survival (or ruin) probability calculation in exchange for a few assumptions on the random variables which generate the renewal risk model. More precisely, we show the expressibility of the distribution function $$\mathbb{P}\left(\sup_{n\geqslant1}\sum_{i=1}^{n}(X_i-c\theta_i)<u\right),\,u\in\mathbb{N}_0$$ via the roots of the probability generating function $G_{X-c\theta}(s)=1$, the expectation $\mathbb{E}(X-c\theta)$, and the probability mass function of $X-c\theta$. We assume that the random variables $X_1,\,X_2,\,\ldots$ and $c\theta_1,\,c\theta_2,\,\ldots$ are independent copies of $X$ and $c\theta$ respectively, $c>0$, $X$ and $c\theta$ are independent non-negative and integer-valued, and the support of $\theta$ is finite. We give few numerical outputs of the proven theoretical statements when the mentioned random variables admit some particular distributions.