Quasi-stationary distributions and convergence to quasi-stationarity of birth-death processes

Abstract

For a birth–death process (X(t), ) on the state space {−1, 0, 1, ·· ·}, where −1 is an absorbing state which is reached with certainty and {0, 1, ·· ·} is an irreducible class, we address and solve three problems. First, we determine the set of quasi-stationary distributions of the process, that is, the set of initial distributions which are such that the distribution of X(t), conditioned on non-absorption up to time t, is independent of t. Secondly, we determine the quasi-limiting distribution of X(t), that is, the limit as t→∞ of the distribution of X(t), conditioned on non-absorption up to time t, for any initial distribution with finite support. Thirdly, we determine the rate of convergence of the transition probabilities of X(t), conditioned on non-absorption up to time t, to their limits. Some examples conclude the paper. Our main tools are the spectral representation for the transition probabilities of a birth–death process and a duality concept for birth–death processes.

Uniform Conditional Ergodicity and Intrinsic Ultracontractivity

“…For an example see e.g. [6] or [34]. The question which is of particular interest is under which conditions on the initial distribution ϑ the limits exist and are equal to a given quasi-stationary distribution (not necessarily the Yaglom limit).…”

Decay parameter and related properties of 2-type branching processes

“…The deep relationship between invariant measures and quasi-stationary distributions has been successfully revealed by the important work of Van Doorn [14] , and Nair and Pollett [11] .…”

“…The decay parameter, on the other hand, was developed by Kingman in early sixties of the last century. Beginning with the pioneer and remarkable work of Kingman [2] and Vere-Jones [3] , this extremely useful theory has been flourished due to many important researches including the significant contributions made by Flaspohler [4] , Pollett [5−7] , Darroch and Seneta [8] , Kelly [9] , Kijima [10] , Nair and Pollett [11] , Tweedie [12] , Van Doorn [13,14] and many others.…”

“…14) has a unique solution (u(r), v(r)). Moreover,(i) u(r), v(r) are continuous and strictly decreasing in [0, r * ], with values in [u * , q 1 ] and [v * , q 2 ], respectively.…”

Decay Properties of n-type Markov Branching Processes with Disasters

“…For more recent developments, one can refer to the survey paper of Van Doorn and Pollett [18]. For the related works, see [6,7,[15][16][17]21].…”