Periodicity and limiting distribution

A Markov chain alternates strictly between two states every step. What is its period, and does it have a unique limiting distribution?

Show hints (2)+
  1. Returns happen only at even times.
  2. Periodicity blocks a limit but not a time-average stationary law.

Answer

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Period 2; no limiting distribution (time-average is 1/2,1/21/2,1/2)

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