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Suppose the matrix A gives the rates for an irreducible, continuous-time Markov chain Xt on a finite state space. Suppose the invariant probability measure is T. For i, j let Aij x = -Aii Xi and P = 0. Note that P is the transition matrix for a discrete time Markov chain that mirrors the transitions of the continuous chain whenever it moves (Durrett calls this the embedded chain.) (a) The discrete chain described by P is irreducible: what is its invariant distribution in terms of and A?
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Question Text | Suppose the matrix A gives the rates for an irreducible, continuous-time Markov chain Xt on a finite state space. Suppose the invariant probability measure is T. For i, j let Aij x = -Aii Xi and P = 0. Note that P is the transition matrix for a discrete time Markov chain that mirrors the transitions of the continuous chain whenever it moves (Durrett calls this the embedded chain.)
(a) The discrete chain described by P is irreducible: what is its invariant distribution in terms of and A?
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Topic | All topics |
Subject | Physics |
Class | Class 12 |