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Media type:
E-Article
Title:
Quasi-stationary distributions and convergence to quasi-stationarity of birth-death processes
Contributor:
Van Doorn, Erik A.
imprint:
Cambridge University Press (CUP), 1991
Published in:Advances in Applied Probability
Language:
English
DOI:
10.2307/1427670
ISSN:
0001-8678;
1475-6064
Origination:
Footnote:
Description:
<jats:p>For a birth–death process (<jats:italic>X</jats:italic>(<jats:italic>t</jats:italic>), <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0001867800023880_inline1" />) 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 <jats:italic>X</jats:italic>(<jats:italic>t</jats:italic>), conditioned on non-absorption up to time <jats:italic>t</jats:italic>, is independent of <jats:italic>t.</jats:italic> Secondly, we determine the quasi-limiting distribution of <jats:italic>X</jats:italic>(<jats:italic>t</jats:italic>), that is, the limit as <jats:italic>t</jats:italic>→∞ of the distribution of <jats:italic>X</jats:italic>(<jats:italic>t</jats:italic>), conditioned on non-absorption up to time <jats:italic>t</jats:italic>, for any initial distribution with finite support. Thirdly, we determine the rate of convergence of the transition probabilities of <jats:italic>X</jats:italic>(<jats:italic>t</jats:italic>), conditioned on non-absorption up to time <jats:italic>t</jats:italic>, 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.</jats:p>