Time Reversibility - Stochastic Processes

Stochastic Processes

A formal definition of time-reversibility is stated by Tong in the context of time-series. In general, a Gaussian process is time-reversible. The process defined by a time-series model which represents values as a linear combination of past values and of present and past innovations (see Autoregressive moving average model) is, except for limited special cases, not time-reversible unless the innovations have a normal distribution (in which case the model is a Gaussian process).

A stationary Markov Chain is reversible if the transition matrix {p_ij} and the stationary distribution {π_j} satisfy

for all i and j. Such Markov Chains provide examples of stochastic processes which are time-reversible but non-Gaussian.

Time reversal of numerous classes of stochastic processes have been studied including Lévy processes stochastic networks (Kelly's lemma) birth and death processes Markov chains and piecewise deterministic Markov processes.