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Hitting probabilities for the Greenwood model and relations to near constancy oscillation. (English) Zbl 1391.60214

The paper deals with the Greenwood model. It may be seen as a model for a population in discrete time, where at each time step each living individual, independent of all other individuals, survives with probability \(p \in (0,1)\), and dies otherwise. More formally, the model is given by a Markov chain \((X_k)_{k \in \mathbb{N}_0}\) on \(\mathbb{N}_0\) with transition matrix \(P=(p_{ij})_{i,j \in \mathbb{N}_0}\) given by \[ p_{ij} = \begin{cases} \binom{i}{j} p^j (1-p)^{i-j} & \text{for } j \leq i; \\ 0 & \text{for } j > i. \end{cases} \] The main results of the paper concern the hitting probabilities \[ h(i,j) = \mathbb{P}(X_k = j \text{ for some } k \in \mathbb{N}_0 \mid X_0 = i). \] Clearly, \(h(i,j) = 0\) for all \(j > i\) and \(h(i,0) = 1\) for all \(i \in \mathbb{N}_0\). For the non-trivial hitting probability \(h(i,j)\) with \(0 < j \leq i\), the author derives the explicit expression \[ h(i,j) = (1-p^j) \binom{i}{j} \sum_{k=j}^i \binom{i-j}{k-j} \frac{(-1)^{k-j}}{1-p^k}{(*)} \] by means of calculations involving the Green matrix \(G=\sum_{n \geq 0} P^n\) and the spectral decomposition of \(P\). The author further provides the expression \[ h(i,j) = \binom{i}{j} \mathbb{E}\big[p^{jM_{i-j}}\big], \quad i,j \in \mathbb{N}_0,\; i > j{(**)} \] where \(M_{i-j}\) is the maximum of \(i-j\) i.i.d. random variables \(G_1,\ldots,G_{i-j}\) with \(\mathbb{P}(G_1=k)=(1-p)p^{k-1}\). Two proofs are given for \((**)\), the first one based on direct calculations using \((*)\), the second based on the use of generating functions. Further, the author studies the asymptotic behavior of \(h(i,j)\) as \(i \to \infty\). The quantity \(h(i,j)\) varies with \(i \to \infty\), and is shown to possess infinitely many accumulation points.
Finally, the existence of a Markov chain \(Y\) which is the Siegmund dual of \(X\) is shown, and the hitting probabilities of \(Y\) are investigated.

MSC:

60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
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