In probability theory, a stochastic process is said to have stationary increments if its change only depends on the time span of observation, but not on the time when the observation was started. Many large families of stochastic processes have stationary increments either by definition (e.g. Lévy processes) or by construction (e.g. random walks)

Definition

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A stochastic process   has stationary increments if for all   and  , the distribution of the random variables

 

depends only on   and not on  .[1][2]

Examples

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Having stationary increments is a defining property for many large families of stochastic processes such as the Lévy processes. Being special Lévy processes, both the Wiener process and the Poisson processes have stationary increments. Other families of stochastic processes such as random walks have stationary increments by construction.

An example of a stochastic process with stationary increments that is not a Lévy process is given by  , where the   are independent and identically distributed random variables following a normal distribution with mean zero and variance one. Then the increments   are independent of   as they have a normal distribution with mean zero and variance two. In this special case, the increments are even independent of the duration of observation   itself.

Generalized Definition for Complex Index Sets

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The concept of stationary increments can be generalized to stochastic processes with more complex index sets  . Let   be a stochastic process whose index set   is closed with respect to addition. Then it has stationary increments if for any  , the random variables

 

and

 

have identical distributions. If   it is sufficient to consider  .[1]

References

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  1. ^ a b Klenke, Achim (2008). Probability Theory. Berlin: Springer. p. 190. doi:10.1007/978-1-84800-048-3. ISBN 978-1-84800-047-6.
  2. ^ Kallenberg, Olav (2002). Foundations of Modern Probability (2nd ed.). New York: Springer. p. 290.