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:href="S0001867800023880_inline1" xlink:type="simple" />) 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>