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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 $$ \mathbb{P}\left(\sup_{n\geqslant1}\sum_{i=1}^{n}(X_i-cθ_i)<u\right),\,u\in\mathbb{N}_0 $$ via the roots of the probability generating function $G_{X-cθ}(s)=1$, the expectation $\mathbb{E}(X-cθ)$, and the probability mass function of $X-cθ$. We assume that the random variables $X_1,\,X_2,\,\ldots$ and $cθ_1,\,cθ_2,\,\ldots$ are independent copies of $X$ and $cθ$ respectively, $c>0$, $X$ and $cθ$ are independent non-negative and integer-valued, and the support of $θ$ is finite. We give few numerical outputs of the proven theoretical statements when the mentioned random variables admit some particular distributions.