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 ...
, independent increments are a property of
stochastic processes and
random measure In probability theory, a random measure is a measure-valued random element. Random measures are for example used in the theory of random processes, where they form many important point processes such as Poisson point processes and Cox processes. ...
s. Most of the time, a process or random measure has independent increments by definition, which underlines their importance. Some of the stochastic processes that by definition possess independent increments are 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 ...
, all
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, all
additive process An additive process, in probability theory, is a cadlag, Continuous stochastic process#Continuity in probability, continuous in probability stochastic process with independent increments.
An additive process is the generalization of a Lévy process ...
and the
Poisson point 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 ...
.
Definition for stochastic processes
Let
be a
stochastic process. In most cases,
or
. Then the stochastic process has independent increments if and only if for every
and any choice
with
:
the
random variables
:
are
stochastically independent
Independence is a fundamental notion in probability theory, as in statistics and the theory of stochastic processes. Two events are independent, statistically independent, or stochastically independent if, informally speaking, the occurrence o ...
.
Definition for random measures
A
random measure In probability theory, a random measure is a measure-valued random element. Random measures are for example used in the theory of random processes, where they form many important point processes such as Poisson point processes and Cox processes. ...
has got independent increments if and only if the random variables
are
stochastically independent
Independence is a fundamental notion in probability theory, as in statistics and the theory of stochastic processes. Two events are independent, statistically independent, or stochastically independent if, informally speaking, the occurrence o ...
for every selection of
pairwise disjoint
In mathematics, two sets are said to be disjoint sets if they have no element in common. Equivalently, two disjoint sets are sets whose intersection is the empty set.. For example, and are ''disjoint sets,'' while and are not disjoint. A c ...
measurable sets
and every
.
Independent S-increments
Let
be a random measure on
and define for every bounded measurable set
the random measure
on
as
:
Then
is called a random measure with independent S-increments, if for all bounded sets
and all
the random measures
are independent.
Application
Independent increments are a basic property of many stochastic processes and are often incorporated in their definition. The notion of independent increments and independent S-increments of random measures plays an important role in the characterization of
Poisson point 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 ...
and
infinite divisibility
References
[ ]
[ ]
[ {{cite book , last1=Kallenberg , first1=Olav , author-link1=Olav Kallenberg , year=2017 , title=Random Measures, Theory and Applications, location= Switzerland , publisher=Springer , doi= 10.1007/978-3-319-41598-7, isbn=978-3-319-41596-3, pages=87 ]
Probability theory