Distribution of Shifted Discrete Random Walk and Vandermonde matrices

TL;DR

This paper derives the generating function for the ultimate time survival probability of a shifted discrete random walk with integer-valued steps, providing explicit formulas and examples for various distributions.

Contribution

It introduces a novel generating function approach for the survival probability of shifted discrete random walks and expresses it via polynomial roots, with applications to common distributions.

Findings

Derived explicit generating functions for survival probabilities

Expressed probabilities via roots of specific polynomials

Provided examples for Bernoulli, Geometric, and other distributions

Abstract

In this work we set up the generating function of the ultimate time survival probability $\varphi(u+1)$, where $$\varphi(u)=\mathbb{P}\left(\sup_{n\geqslant 1}\sum_{i=1}^{n}\left(X_i-\kappa\right)<u\right)$$ and $u\in\mathbb{N}_0,\,\kappa\in\mathbb{N}$, 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 $\varphi(u)$ via the roots of certain polynomials. Based on the proven theoretical statements, we give several examples on $\varphi(u)$ and its generating function expressions, when random variables $X_i$ admit Bernoulli, Geometric and some other distributions.