Let (Zt) be a subordinator independent of 0 = U = 1 and let u and v be positive constants. Solutions to the "in law" equation Zu = dUZu+v exist under certain conditions and they have a distribution function which is continuous on the positive reals. A discrete version of this equation is here...

The concept of a limiting conditional age distribution of a continuous time Markov process whose state space is the set of non-negative integers and for which {0} is absorbing is defined as the weak limit as t--[infinity] of the last time before t an associated "return" Markov process exited...

A Markov branching process with instantaneous immigration from the zero state can be constructed so as to be honest and have the non-negative integers as state-space, but the construction requires the branching part to be explosive. We show that a realistic model can be constructed without this...

Let (Nt: t [greater-or-equal, slanted] 0), with N0 = 0, be the counting process generated by independent sojourn-times Xj in states . Interest is focussed on the rate at which (Nt) explodes to infinity when T[infinity] = [Sigma]j[greater-or-equal, slanted]1 Xj [infinity] a.s. Reasonably...

This paper considers a Markov branching process modified to allow decrements which occur randomly at a rate proportional to the population size. The problems considered are (i) Conditions for eventual extinction, (ii) Asymptotic behaviour of the probability of extinction and the mean time to...

It is known that, subject to some technical conditions, the deficit process of an infinitely deep dam with a Markov chain input process and unit withdrawals has a limiting zero-modified geometric distribution. In this paper it is shown that, provided the state space of the input process is...

The stationary conditional quasi-stationary distribution of the linear birth, death and catastrophe process is shown to exist iff the decrement distribution has a finite second order moment, Conditional limit theorems for the population size are found when this moment is infinite and a regular...

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...