Distribution of shifted discrete random walk generated by distinct random variables and applications in ruin theory

Abstract

In this paper, we set up the distribution function $$\phi (u)=\mathbb{P}\left(\underset{n⩾1}{\sup }\sum _{i=1}^n\left(X_i-\kappa \right)<u\right),$$ and the generating function of $\phi (u+1)$, where $u\in {\mathbb{N}}_0$, $\kappa \in \mathbb{N}$, the random walk $\left\{{\sum }_{i=1}^n{X}_i,n\in \mathbb{N}\right\},$ consists of $N\in \mathbb{N}$ periodically occurring distributions, and the integer-valued and non-negative random variables $X_1,X_2,\dots$ are independent. This research generalizes two recent works where $\{\kappa =1,N\in \mathbb{N}\}$ and $\{\kappa \in \mathbb{N},N=1\}$ were considered respectively. The provided sequence of sums $\left\{{\sum }_{i=1}^n\left(X_i-\kappa \right),n\in \mathbb{N}\right\}$ generates so-called multi-seasonal discrete-time risk model with arbitrary natural premium and its known distribution enables to calculate the ultimate time ruin probability $1-\phi (u)$ or survival probability $\phi (u)$. Verifying obtained theoretical statements we demonstrate several computational examples for survival probability $\phi (u)$ and its generating function when $\{\kappa =2,N=2\}$, $\{\kappa =3,N=2\}$, $\{\kappa =5,N=10\}$ and $X_i$ admits Poisson and some other distributions. We also conjecture the non-singularity of certain matrices.

References in corpus (1)

• Distribution of Shifted Discrete Random Walk and Vandermonde matrices