On a Markov chain model for population growth subject to rare catastrophic events

We consider a Markov chain model for population growth subject to rare catastrophic events. In this model, the moves of the process are getting algebraically rare (as from x−λ) when the process visits large heights x, and given a move occurs and the height is large, the chain grows by one unit with large probability or undergoes a rare catastrophic event with small complementary probability ∼ γ/x. We assume pure reflection at the origin. This chain is irreducible and aperiodic; it is always recurrent, either positive or null recurrent. Estimates are obtained for first-return time probabilities to the origin (excursion length), eventual return (contact) probability and excursion height. All exhibit power-law decay in some range of the parameters (γ, λ). We show a scaling relationship between heights and lengths of the excursions. From this, the mean and median of both the empirical average and sample maximum are shown to grow algebraically with exponents being identified in terms of (γ, λ).