The limiting distribution for the infinitely deep dam with a Markovian input

Abstract

The paper outlines a case for taking greater interest in the bottomless, or infinitely deep, dam model in Hydrology. It then shows that for such a model with unit withdrawals and an ergodic Markov chain input process the limiting distribution of depletion, when this exists, is a zero modified geometric distribution. This result generalises the well known result for independent inputs. The technical conditions required for the proof are satisfied for finite state space input processes and are shown to be satisfied by certain infinite state space input processes. These include as special cases examples which have a negative binomial limiting input distribution.