Mathematics · Markov Chains
Consider an irreducible Markov chain with a finite state space S. Let P = ((pij)) be its transition probability matrix and let pij(n) denote the n-step transition probability from state i to state j. Let αij = limn → ∞ 1n ∑m=1n pij(m) for i, j ∈ S. Which of the following statements is necessarily true?
Verified Solution4 OptionsUpdated: October 2026👁️1 View
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- 1The sequence pij(n) always converges to αij as n → ∞ for all i, j ∈ S
- 2αij > 0 for all i, j ∈ S✓
- 3∑i ∈ S αij = 1 for all j ∈ S
- 4αij = αkj for all i, j, k ∈ S
Correct answer
αij > 0 for all i, j ∈ S
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