On $2\times 2$ determinants originating from survival probabilities in homogeneous discrete time risk model

Abstract

We analyze $2\times 2$ Hankel-like determinants $D_n$ that arise in the initial values problem for the ultimate time survival probability $\phi (u)$ in a homogeneous discrete time risk model $W(n)=u+\kappa n+{\sum }_{i=1}^n{Z}_i$, where $Z_i$ are positive integer valued i.i.d. random claims, the initial surplus $u\in {\mathbb{N}}_0$ and the income rate $\kappa =2$. We prove the asymptotic version of a recent conjecture on the non--vanishing and monotonicity of $D_n$ and derive explicit formulas for the initial values $\phi (0)$, $\phi (1)$ of a recurrence that yields survival probabilities. In cases when $Z_i$ are Bernoulli or Geometrically distributed, the conjecture on $D_n$ is shown to hold for all $n\in {\mathbb{N}}_0$. Additionally, a generating function $\Xi (s)$ for ultimate survival probabilities $\phi (u)$ is derived.

arXiv commentV3 revision of a submitted version with incorporated Referee's remarks. 27 pages, 1 figure; minor changes since v2(switched from "t" to "s" in generating functions; corrections of some misprints; additions to acknowledgments, etc.)