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) A stochastic process
, the distribution of the random variables depends only on
[1][2] Having stationary increments is a defining property for many large families of stochastic processes such as the Lévy processes.
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
are independent and identically distributed random variables following a normal distribution with mean zero and variance one.
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
The concept of stationary increments can be generalized to stochastic processes with more complex index sets
be a stochastic process whose index set
is closed with respect to addition.
, the random variables and have identical distributions.