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##### (searched for: doi:10.15559/22-vmsta209)
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Published: 1 January 2022
Journal: AIMS Mathematics
AIMS Mathematics, Volume 8, pp 5181-5199; https://doi.org/10.3934/math.2023260

Abstract:
In this work, we set up the generating function of the ultimate time survival probability $\varphi(u+1)$, where \begin{document}$\varphi(u) = \mathbb{P}\left(\sup\limits_{n\geqslant 1}\sum\limits_{i = 1}^{n}\left(X_i- \kappa\right)<u\right),$\end{document} $u\in\mathbb{N}_0, \, \kappa\in\mathbb{N}$ 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. The probability $\varphi(u)$ means that the stochastic process \begin{document}$u+ \kappa n-\sum\limits_{i = 1}^{n}X_i$\end{document} 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 $\varphi(u)$ and its generating function expressions, when random variables $X_i$ admit Bernoulli, geometric and some other distributions. Page of 1
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