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

We analyze $2\times 2$ Hankel-like determinants $D_n$ that arise in the initial values problem for the ultimate time survival probability $φ(u)$ in a homogeneous discrete time risk model $W(n)=u+κn+\sum_{i=1}^nZ_i$, where $Z_i$ are positive integer valued i.i.d. random claims, the initial surplus $u \in \mathbb{N}_0$ and the income rate $κ=2$. 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 $φ(0)$, $φ(1)$ 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 $Ξ(s)$ for ultimate survival probabilities $φ(u)$ is derived.

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