Statistics Seminars: Quasistationary distributions for continuous-time Markov chains and bounds for the decay parameter of a birth-death process
7 April 2006 00:00 in CM107"Quasi stationarity is a notion used to describe the behaviour of processes that eventually die out, but display stationary-like behaviour over any reasonable time-scale. For example, a threatened species may survive for extended periods before becoming extinct; a telecommunications network may fluctuate between congested and uncongested states without any apparent change in demand, and stay in each state for long periods; and a chemical system where one species can become depleted (and thus stop the reaction) may settle to a stable equilibrium.
I will summarise known results in this area and look at some of the many avenues available for further research. I will illustrate these results with reference to a particular class of auto-catalytic chemical reactions.
Central to the theory of quasistationary distributions is a quantity known as the decay parameter, which describes the rate of exponential decay of the transition probabilities of an absorbing Markov chain. Despite its importance, the decay parameter is notoriously difficult to evaluate or even approximate.
I will outline a non-standard characterisation of the decay parameter and indicate how this leads to explicit bounds for the decay parameter of a general birth-death process. An immediate corollary is a necessary and sufficient condition for positivity of the decay parameter; which I will illustrate with several examples. This is joint work with Hanjun Zhang and Phil Pollett."
CM103 @ 11:00
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