Discrete-time stochastic process

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In probability theory and statistics, a discrete-time stochastic process is a stochastic process for which the index variable takes a discrete set of values, as contrasted with a continuous-time process for which the index variable takes values in a continuous range. An alternative terminology uses discrete parameter as being more inclusive.[1]

A more restricted class of processes are those with discrete time and discrete state space. The apparently simpler terms "discrete process" or "discontinuous process" may cause confusion with processes having continuous time and discrete state space.[2] Given the possible confusion, caution is needed.[2]


Examples of discrete-time stochastic processes are random walks and branching processes, for which the state space may be either continuous or discrete. Important examples of discrete time and continuous state space processes are models conventionally used in time series analysis: for example, the autoregressive, vector autoregressive, moving average, ARMA, ARIMA and ARCH models.

See also[edit]


  1. ^ Parzen, E. (1962) Stochastic Processes, Holden-Day. ISBN 0-8162-6664-6 (page 7)
  2. ^ a b Dodge, Y. (2006) The Oxford Dictionary of Statistical Terms, OUP. ISBN 0-19-920613-9 (Entry for "continuous process")