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Symmetry. The distinct pair i, j of states of a Markov chain is called symmetric if P(Tj < Ti | Xo = i) = P(Ti < Tj | Xo = j), where Ti = min {n ≥ 1: Xn=i}. Show that, if Xo = i and i, j is symmetric, the expected number of visits to j before the chain revisits i i is 1.

Symmetry. The distinct pair i, j of states of a Markov chain is called symmetric if P(Tj < Ti | Xo = i) = P(Ti < Tj | Xo = j), where Ti = min {n ≥ 1: Xn=i}. Show that, if Xo = i and i, j is symmetric, the expected number of visits to j before the chain revisits i i is 1.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter3: Matrices
Section3.7: Applications
Problem 2EQ
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Symmetry. The distinct pair i, j of states of a Markov chain is called symmetric if P(Tj < Ti | Xo = i) = P(Ti < Tj | Xo = j), where Ti = min {n ≥ 1: Xn=i}. Show that, if Xo = i and i, j is symmetric, the expected number of visits to j before the chain revisits i i is 1.
Transcribed Image Text:Symmetry. The distinct pair i, j of states of a Markov chain is called symmetric if P(Tj < Ti | Xo = i) = P(Ti < Tj | Xo = j), where Ti = min {n ≥ 1: Xn=i}. Show that, if Xo = i and i, j is symmetric, the expected number of visits to j before the chain revisits i i is 1.
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