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 $$\mathbb{P}(\underset{n⩾1}{\sup }\sum _{i=1}^n(X_i-c{\theta }_i)<u),u\in {\mathbb{N}}_0$$ 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 $X_1,X_2,\dots$ and $c{\theta }_1,c{\theta }_2,\dots$ 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.