Invariant measures and positive recurrence

An invariant measure assigns a nonnegative finite weight to each state and satisfies incoming balance at every state. Its total mass may be infinite. A stationary distribution is an invariant measure whose weights sum to one. If the chain starts with that law, its distribution stays the same at every time, even though individual paths continue moving. Multiplying an invariant measure by a positive constant preserves invariance, so uniqueness of an unnormalized invariant measure is normally understood up to a scalar under the relevant recurrence hypotheses.

For a finite irreducible chain, a stationary probability exists, is unique and is positive on every state. For an irreducible countable chain, existence of a stationary probability is equivalent to positive recurrence. In that case it is unique and assigns positive mass everywhere. An irreducible recurrent chain also possesses a nonzero statewise-finite invariant measure unique up to scaling, but in the null recurrent case its total mass is infinite. Transient chains may have invariant measures too, so merely writing down one does not certify recurrence or equilibrium.

For an irreducible positive recurrent countable chain, the stationary probability of x equals the reciprocal of the expected positive return time to x. A cycle beginning at x and ending immediately before its next return contains one visit to x. More generally, the expected number of visits to y during that cycle, divided by the expected cycle length, produces the stationary mass of y. This regenerative interpretation explains both normalization and the obstruction in null recurrence: an infinite mean cycle cannot yield a probability by this finite denominator.

The return formula does not require aperiodicity. A deterministic cycle of three states has stationary mass one third at each state and mean return length three, yet its distribution from one state never approaches that stationary law along all times. For an irreducible, positive recurrent, aperiodic countable chain, the distribution from each fixed initial state does converge in total variation to the stationary law. This is pointwise in the initial state; it need not imply a finite worst-case mixing time when the state space is infinite and starts arbitrarily far away remain far from equilibrium.

For a nearest-neighbor chain on the nonnegative integers, balancing adjacent flows gives a recursive construction of invariant weights. If the upward probability is p and the downward probability away from zero is q, both strictly positive, adjacent weights have ratio p/q, with any boundary holding chosen to make rows sum to one. A ratio below one gives a summable geometric tail. A ratio of one gives constant weights, and a ratio above one gives growing weights. This simple tail calculation can reveal whether a candidate stationary probability exists before solving a large linear system.

Failure of summability distinguishes absence of equilibrium from a missing normalization constant, but it does not by itself distinguish null recurrence from transience. For the reflecting nearest-neighbor chain with equal upward and downward probabilities one half, recurrence follows from the fair-walk hitting calculation and the invariant counting measure has infinite mass, so the chain is null recurrent. With outward drift it is transient. A truncated version of either chain has a stationary law because it is finite; those normalized finite laws can send their mass toward the truncation boundary instead of converging to an infinite-space stationary probability.