Editing Independent increments

In [[probability theory]], '''independent increments''' are a property of [[stochastic process]]es and [[random measure]]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]], all [[Lévy process]]es, all [[additive process]]<ref>{{cite book |last1=Sato |first1=Ken-Ito |title=Lévy processes and infinitely divisible distributions |date=1999 |pages=31-68|publisher=Cambridge University Press |isbn=9780521553025}}</ref>
and the [[Poisson point process]].

== Definition for stochastic processes ==
Let <math> (X_t)_{t \in T} </math> be a [[stochastic process]]. In most cases, <math> T= \N </math> or <math> T=\R^+ </math>. Then the stochastic process has independent increments if and only if for every <math> m \in \N </math> and any choice <math> t_0, t_1, t_2, \dots,t_{m-1}, t_m \in T</math> with
:<math>  t_0 < t_1 < t_2< \dots < t_m </math>

the [[random variables]]
:<math> (X_{t_1}-X_{t_0}),(X_{t_2}-X_{t_1}), \dots, (X_{t_m}-X_{t_{m-1}} )</math>

are [[Independence (probability theory)|stochastically independent]].<ref name="Klenke190" />

== Definition for random measures ==
A [[random measure]] <math> \xi </math> has got independent increments if and only if the random variables <math>\xi(B_1), \xi(B_2), \dots, \xi(B_m) </math> are [[Independence (probability theory)|stochastically independent]] for every selection of [[pairwise disjoint]] measurable sets <math> B_1, B_2, \dots, B_m </math> and every <math> m \in \N </math>. <ref name="Klenke527" />

== Independent S-increments ==
Let <math> \xi </math> be a random measure on <math> S \times T </math> and define for every bounded measurable set <math> B </math> the random measure <math> \xi_B </math> on <math> T </math> as
:<math> \xi_B(\cdot):= \xi(B \times \cdot ) </math>

Then <math> \xi </math> is called a random measure with '''independent S-increments''', if for all bounded sets <math> B_1, B_2, \dots, B_n </math> and all <math> n \in \N </math> the random measures <math> \xi_{B_1},\xi_{B_2}, \dots, \xi_{B_n}</math> are independent.<ref name="Kallenberg87" />

== 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]] and [[infinite divisibility]].

== References ==
<references>
<ref name="Klenke527" >{{cite book |last1=Klenke |first1=Achim |year=2008  |title=Probability Theory |location=Berlin |publisher=Springer |doi=10.1007/978-1-84800-048-3 |isbn=978-1-84800-047-6|pages= 527 }} </ref>
<ref name="Klenke190" >{{cite book |last1=Klenke |first1=Achim |year=2008  |title=Probability Theory |location=Berlin |publisher=Springer |doi=10.1007/978-1-84800-048-3 |isbn=978-1-84800-047-6|pages= 190 }} </ref>
<ref name="Kallenberg87" > {{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}} </ref>
</references>

[[Category:Probability theory]]