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

(X_t)_

be a stochastic process. In most cases,

T= \N

T=\R^+

. Then the stochastic process has independent increments if and only if for every

m \in \N

and any choice

t_0, t_1, t_2, \dots,t_, t_m \in T

with :

t_0 < t_1 < t_2< \dots < t_m

the random variables :

(X_-X_),(X_-X_), \dots, (X_-X_ )

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

\xi

has got independent increments if and only if the random variables

\xi(B_1), \xi(B_2), \dots, \xi(B_m)

are

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

B_1, B_2, \dots, B_m

and every

m \in \N

Independent S-increments

Let

\xi

be a random measure on

S \times T

and define for every bounded measurable set

B

the random measure

\xi_B

T

as :

\xi_B(\cdot):= \xi(B \times \cdot )

Then

\xi

is called a random measure with independent S-increments, if for all bounded sets

B_1, B_2, \dots, B_n

and all

n \in \N

the random measures

\xi_,\xi_, \dots, \xi_

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

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