The age of a Markov process

Anthony G. Pakes

Stochastic Processes and their Applications, 1979, vol. 8, issue 3, 277-303

Abstract: 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 from {0} conditional on the state, j, of this process at t. It is shown that this limit exists and is non-defective if the return process is [rho]-recurrent and satisfies the strong ratio limit property. As a preliminary to the proof of the main results some general results are established on the representation of the [rho]-invariant measure and function of a Markov process. The conditions of the main results are shown to be satisfied by the return process constructed from a Markov branching process and by birth and death processes. Finally, a number of limit theorems for the limiting age as j-->[infinity] are given.

Keywords: Markov; process; [rho]-classification; Markov; branching; process; limit; theorems; limiting; age; strong; ratio; limit; property; birth; and; death; process; Green; function (search for similar items in EconPapers)
Date: 1979
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