Ergodic properties of nonhomogeneous, continuous-time Markov chains

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1984
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Johnson, Jean
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Abstract

A continuous-time Markov chain on a discrete, possibly infinite, state space with probability transition matrices P(s,t) has probability p(,ij)(s,t) of going to state j at time t if it is in state i at time s. This chain can often be defined alternately by its intensity matrix;(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI);For homogeneous chains Q(t) is independent of t. This dissertation considers nonhomogeneous chains for which Q(t) does change with t;Two theorems are given which establish sufficient conditions for strong ergodicity of nonhomogeneous Markov chains. First, if the chain displays loss of memory (weak ergodicity) and if the eigenvectors (psi)(t) of Q(t) associated with eigenvalue zero converge to a probability distribution (psi) at a rate such that;(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI);then Q(t) defines a strongly ergodic chain with (psi) as its long-run distribution. Second, if (PARLL)(Q)t - Q(PARLL) (--->) 0 as t (--->) (INFIN) and Q is the intensity matrix of a strongly ergodic, homogeneous chain, then the nonhomogeneous chain defined by Q(t) is also strongly ergodic and has the same long-run distribution. A theorem on the rate of convergence of strongly ergodic chains is also presented;Nonhomogeneous chains defined by;(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI);or;(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI);are restudied. The Q(,k)'s commute, are bounded in norm by 2q(,k) < (INFIN), and are the intensity matrices of homogeneous, continuous-time chains. Properties such as irreducibility, ergodicity and strong ergodicity are studied in terms of the corresponding properties in the homogeneous, discrete-time chains given by the probability transition matrices (')P(,k) = I + (1/q(,k))Q(,k). Since homogeneous, discrete-time chains have been studied extensively, these properties are well known for the (')P(,k)'s. The special case where Q(t) = tC + Q is considered separately since this corresponds to a continuous-time version of the discrete constant-causative chain.

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Sun Jan 01 00:00:00 UTC 1984