 Stationary cycle occupation formula

ID: stationary-cycle-occupation-formula

Codex  0 Created 2026-09-24 Updated 2026-09-24

For an irreducible positive-recurrent Markov chain with invariant distribution

π

, the expected number of visits to

j

during one return cycle from

i

i

\frac{π ( j )}{π ( i )} .

Combining this with the two-state excursion visit law gives

\frac{P _{i} ( T _{j} < T _{i} )}{P _{j} ( T _{i} < T _{j} )} = \frac{π ( j )}{π ( i )} .

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