Period and time averages
Section of Classes, recurrence, and reversibility, part of Stochastic processes
Period and time averages
| Type | Section |
| Taught in | Classes, recurrence, and reversibility |
| Part | Stochastic processes |
A deterministic three-cycle has period 3. Started at state 1, its distribution cycles through the three point masses. Over each complete set of three steps it spends exactly one third of its time in each state.
For a finite irreducible chain, the empirical frequency of visits to j along one path converges to \pi_j, even when the one-time probabilities p_{ij}^{(n)} oscillate. Two limits are in play. One is a time average along a path. The other is the distribution at one distant time. The path average is the law of large numbers for a dependent sequence, already used on Markov chains to justify estimating \pi by a long run. Period blocks the second limit and leaves the first intact.
In a finite irreducible chain the stationary weight is the reciprocal mean return time,
\pi_i=\frac{1}{\mathbb{E}_i T_i^+}.
A state that is visited often has a short mean return. On the graph walk, a high-degree vertex carries more stationary mass and so has the shorter mean return. The identity is a check on a stationary distribution that was solved another way.
Used in
See also
Recurrence and period · Finite irreducible chains · Detailed balance · Communicating classes · Return and period · Global balance and detailed balance · Two families with an explicit stationary law · When a stationary law fails to be unique