  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.
