Distribution of Shifted Discrete Random Walk and Vandermonde matrices

Abstract

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

References in corpus (2)

• Ruin probability for the bi-seasonal discrete time risk model with dependent claims
• Multi seasonal discrete time risk model revisited

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• On the exact survival probability by setting discrete random variables in E. Sparre Andersen's model
• The limit law of certain discrete multivariate distributions
• Distribution of shifted discrete random walk generated by distinct random variables and applications in ruin theory