returntime

Return time, occasionally written as returntime, is the duration between successive visits to a given state or region in a stochastic process or dynamical system. It is typically modeled as a random variable T, with T_i denoting the time to return to state i after leaving it (the first return time). In discrete-time Markov chains, T_i is defined as T_i = inf{ n ≥ 1 : X_n = i } when the process starts in state i. In a finite, irreducible, positive recurrent chain, the expected return time satisfies E_i[T_i] = 1/π_i, where π is the chain’s stationary distribution; consequently, π_i equals the long-run proportion of visits to i.

In continuous-time settings, analogous relations hold for return to a state or set, with rates replacing probabilities. In ergodic theory and dynamical systems, Poincaré recurrence asserts that almost every orbit returns arbitrarily close to its initial position in a finite measure-preserving system. A related result, Kac’s lemma, states that the mean return time to a measurable set A is 1/μ(A) for ergodic, finite-measure systems, linking recurrence to the size of A.

The distribution of return times can vary across systems: some yield geometric or exponential first-return distributions, while others exhibit heavy tails. Return times are used in reliability analysis, queuing theory, ecological modeling, and statistical physics to assess system longevity, mixing properties, and temporal patterns.

See also: Poincaré recurrence, Kac’s lemma, first return time, renewal theory, stationary distribution.