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

► Markov chain for population growth subject to rare catastrophic events. ► Excursion height and length, eventual return probability to origin, time to failure. ► Growth of the mean and median of both the empirical average and sample maximum. ► Based on a scaling relationship between excursions heights and lengths. ► Population growth is slow and algebraic in the number of steps.