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Recurrence Times of Stochastic Processes (also Hitting, Waiting, and First-Passage Times)

08 Dec 2018 11:37

The recurrence time of a state or a finite trajectory is simply how long one must wait to revisit the state, or re-traverse that trajectory. One can learn a lot about a stochastic process by understanding its recurrence times. For instance, Mark Kac proved a very beautiful theorem which says that, for a stationary, discrete-valued stochastic process, the expected recurrence time of a finite trajectory is just the reciprocal of the probability of encountering the trajectory in the first place. This suggests a very simple way to estimate the probability distribution of trajectories. Similarly, one can use the recurrence times to estimate the entropy rate.

Vague, Kac-inspired question: Clearly, if you had a function which gave you the expected recurrence time of an arbitrary finite trajectory, you'd have a function which also gave you all the finite-dimensional marginal distributions of the generating process. But how does one express the higher moments of the recurrence times (the variance, for starters), and might there be some way of trading off knowing more about the higher moments of shorter trajectories for knowing more first moment of longer trajectories? (Getting first moments of long trajectories from first moments of short ones would seem to imply some sort of [conditional] independence.)

See also: Ergodic Theory; Estimating Entropies and Informations; Information Theory; Friedrich Nietzsche; Point Processes; Stochastic Processes; Time Series

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