Ergodicity, weak mixing, and mixing

On a probability-preserving system, ergodicity means that every event invariant modulo null sets has probability zero or one. An equivalent correlation statement says that, for every pair of measurable events, the Cesàro average of their overlap probabilities converges to the product of their probabilities. The deviations can have different signs and cancel in this average. This criterion follows by applying the mean ergodic theorem to an event indicator and pairing with the other indicator. It does not require the individual overlap probabilities to converge.

A single finite cycle with uniform probability is ergodic: an invariant subset must be a union of full cycles, leaving only the empty set or the whole space. Nevertheless any two singleton events have periodic overlap probabilities. This example is already enough to distinguish exploration in time averages from loss of temporal memory. Ergodicity is always relative to the chosen invariant measure; assigning positive mass to two disjoint cycles creates a nontrivial invariant event and destroys whole-space ergodicity.

Weak mixing requires the Cesàro mean of the absolute correlation deviations to tend to zero for every pair of measurable events. Strong mixing requires the deviation itself to tend to zero for every pair. Boundedness of correlations gives the implication from strong mixing to weak mixing, and the ordinary triangle inequality gives the implication from weak mixing to ergodicity. Neither name should be inferred just from the existence of an invariant measure or the convergence of one selected observable's time average.

These notions have equivalent bounded-observable formulations obtained by approximating measurable functions by simple functions. For invertible probability systems, weak mixing is also equivalent to ergodicity of the product transformation on two independent copies, and to absence of nonconstant L2 eigenfunctions. These are useful structural tests, but they do not make weak mixing identical to strong mixing. Examples of weakly mixing systems that are not strongly mixing exist; constructing them takes more work than the simple periodic and independent examples used here.

A stationary Markov chain on a two-sided sequence space defines an invertible probability-preserving shift. Its event correlations can be calculated from a transition matrix, but the stationary initial law must be included. A transition instrument starting from state zero gives conditional transition probabilities instead of a stationary joint law. Multiplying its probability of returning to zero by the stationary mass of zero produces the two-time stationary joint probability. Subtracting the squared stationary mass then gives the indicator correlation.

For an irreducible aperiodic finite chain, transition probabilities converge to the stationary law, and the corresponding stationary shift is strongly mixing. First verify separated cylinder events using transition convergence, then approximate general events by cylinders. A periodic irreducible chain can be stationary and its shift ergodic while its correlations oscillate. Thus the period obstruction is not a failure of stationarity or of long-run component exploration; it specifically obstructs the decorrelation limit required for mixing.