Show that an ergodic markov chain with m states

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(Proof of Theorem 4.2.11) (a) Show that an ergodic Markov chain with M states must contain a cycle with τ <>M states. Hint: Use ergodicity to show that the smallest cycle cannot contain M states.

(b) Let .e be a fixed state on this cycle of length τ . Let (m) be the set of states accessible from .e in steps. Show that for each ≥ 1, (m) ⊆ (τ ). Hint: For any given state ∈ (m), show how to construct a walk of τ steps from .e to from the assumed walk of steps.

(c) Define (0) to be the singleton set {.e} and show that T    (0) ⊆ T    (τ ) ⊆ T    (2τ ) ⊆ ··· ⊆ T    (nτ ) ⊆ ··· .

(d) Show that if one of the inclusions above is satisfied with equality, then all subsequent inclusions are satisfied with equality. Show from this that at most the first - 1 inclusions can be satisfied with strict inequality and that T    (nτ ) = T     ((- 1)τ ) for all ≥ - 1.

(e) Show that all states are included in ((- 1)τ ).

(f) Show that P(M-1)2+1 0 for all ij.

Text Book: Stochastic Processes: Theory for Applications By Robert G. Gallager.

Reference no: EM13909630

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