On $\mathbf{2\times2}$ determinants originating from survival probabilities in homogeneous discrete time risk model

TL;DR

This paper investigates the properties of specific determinants related to survival probabilities in a discrete-time risk model, proving conjectures about their behavior and deriving formulas and generating functions for survival probabilities.

Contribution

It proves the asymptotic non-vanishing and monotonicity of $D_n$, confirms conjectures for Bernoulli and Geometric claims, and derives explicit formulas and generating functions for survival probabilities.

Findings

Proved asymptotic non-vanishing and monotonicity of $D_n$.

Confirmed conjecture for Bernoulli and Geometric claim distributions.

Derived explicit formulas for initial survival probabilities and a generating function.

Abstract

We analyze $2\times 2$ Hankel-like determinants $D_n$ that arise in the initial values problem for the ultimate time survival probability $\varphi(u)$ in a homogeneous discrete time risk model $W(n)=u+\kappa n+\sum_{i=1}^nZ_i$, 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 $\varphi(0)$, $\varphi(1)$ 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 $\varphi(u)$ is derived.