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 {pij} and the stationary distribution {πj} 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.
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“The vast results obtained by Science are won by no mystical faculties, by no mental processes other than those which are practiced by every one of us, in the humblest and meanest affairs of life. A detective policeman discovers a burglar from the marks made by his shoe, by a mental process identical with that by which Cuvier restored the extinct animals of Montmartre from fragments of their bones.”
—Thomas Henry Huxley (1825–95)