Stationary increments
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. A stochastic process formula_1 has stationary increments if for all formula_2 and formula_3, the distribution of the random variables depends only on formula_5 and not on formula_6. Examples. 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 formula_7, where the formula_8 are independent and identically distributed random variables following a normal distribution with mean zero and variance one. Then the increments formula_9 are independent of formula_6 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 formula_5 itself. Generalized Definition for Complex Index Sets. The concept of stationary increments can be generalized to stochastic processes with more complex index sets formula_12. Let formula_13 be a stochastic process whose index set formula_14 is closed with respect to addition. Then it has stationary increments if for any formula_15, the random variables and have identical distributions. If formula_18 it is sufficient to consider formula_19.