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If you have an irreducible Markov chain with transition matrix $P$, and $p(j,j) > 0$ for all $j$, why are all states aperiodic?

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The question says p(j,j) > 0 (same j) and not p(i,j) > 0 for every i,j. It means that if you had a positive probability of going from i to j in n steps (for some n), you can always add 1 to the number of steps, and get a positive probability of going from i to j in n+1 steps (simply by multiplying the transition probability by p(j,j) which we know is positive). We also know that the chain is irreducible, so for every i,j there is at least one n such that going from i to j in n steps has a positive probability. Because you can always add 1 to this n, the greatest common divisor of all such n's must be 1. This is the definition of an aperiodic Markov chain.

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