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)
Full Text: