In
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 ...
s)
Definition
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 ...
has stationary increments if for all
and
, the distribution of the random variables
:
depends only on
and not on
.
Examples
Having stationary increments is a defining property for many large families of stochastic processes such as the
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. 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
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 ...
s 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
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
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
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
.
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