probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...

, a

stochastic process In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appea ...

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 process In probability theory, a Lévy process, named after the French mathematician Paul Lévy, is a stochastic process with independent, stationary increments: it represents the motion of a point whose successive displacements are random, in which disp ...

es) or by construction (e.g.

random walk In mathematics, a random walk is a random process that describes a path that consists of a succession of random steps on some mathematical space. An elementary example of a random walk is the random walk on the integer number line \mathbb Z ...

Definition

X=(X_t)_

has stationary increments if for all

t \geq 0

and

h > 0

, the distribution of the random variables :

Y_:=X_ -X_t

depends only on

h

and not on

t

Examples

Having stationary increments is a defining property for many large families of stochastic processes such as the

es. Being special Lévy processes, both the

Wiener process In mathematics, the Wiener process is a real-valued continuous-time stochastic process named in honor of American mathematician Norbert Wiener for his investigations on the mathematical properties of the one-dimensional Brownian motion. It is o ...

and the

Poisson process In probability, statistics and related fields, a Poisson point process is a type of random mathematical object that consists of points randomly located on a mathematical space with the essential feature that the points occur independently of one ...

es have stationary increments. Other families of stochastic processes such as

s have stationary increments by construction. An example of a stochastic process with stationary increments that is not a Lévy process is given by

X=(X_t)

, where the

X_t

are

independent and identically distributed random variables In probability theory and statistics, a collection of random variables is independent and identically distributed if each random variable has the same probability distribution as the others and all are mutually independent. This property is us ...

following a

normal distribution In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is : f(x) = \frac e^ The parameter \mu ...

with mean zero and variance one. Then the increments

Y_

are independent of

t

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

h

itself.

Generalized Definition for Complex Index Sets

The concept of stationary increments can be generalized to stochastic processes with more complex index sets

T

. Let

X=(X_t)_

be a stochastic process whose index set

T \subset \R

is closed with respect to addition. Then it has stationary increments if for any

p,q,r \in T

, the random variables :

Y_1=X_-X_

and :

Y_2=X_-X_

have identical distributions. If

0 \in T

it is sufficient to consider

r=0

References

{{cite book , last1=Kallenberg , first1=Olav , author-link1=Olav Kallenberg , year=2002 , title=Foundations of Modern Probability, location= New York , publisher=Springer , edition=2nd , page=290 Stochastic processes