Exact expression of ultimate time survival probability in homogeneous discrete-time risk model

In this work, we set up the generating function of the ultimate time survival probability $\phi (u+1)$, where

$$\phi (u)=\mathbb{P}\left(\underset{n⩾1}{sup}\sum _{i=1}^n\left(X_i-\kappa \right)<u\right),$$

$u\in {\mathbb{N}}_0,\kappa \in \mathbb{N}$ and the random walk $\left\{\sum _{i=1}^n{X}_i,n\in \mathbb{N}\right\}$ consists of independent and identically distributed random variables $X_i$, which are non-negative and integer-valued. We also give expressions of $\phi (u)$ via the roots of certain polynomials. The probability $\phi (u)$ means that the stochastic process

$$u+\kappa n-\sum _{i=1}^n{X}_i$$

is positive for all $n\in \mathbb{N}$, where a certain growth is illustrated by the deterministic part $u+\kappa n$ and decrease is given by the subtracted random part $\sum _{i=1}^n{X}_i$. Based on the proven theoretical statements, we give several examples of $\phi (u)$ and its generating function expressions, when random variables $X_i$ admit Bernoulli, geometric and some other distributions.