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Independent increments

In probability theory, independent increments are a property of stochastic processes and random measures. 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 processes and the Poisson point process.

Definition for stochastic processesEdit

Let   be a stochastic process. In most cases,   or  . Then the stochastic process has independent increments iff for every   and any choice   with

 

the random variables

 

are stochastically independent.[1]

Definition for random measuresEdit

A random measure   has got independent increments iff the random variables   are stochastically independent for every selection of pairwise disjoint measurable sets   and every  . [2]

Independent S-incrementsEdit

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.[3]

ApplicationEdit

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

ReferencesEdit

  1. ^ Klenke, Achim (2008). Probability Theory. Berlin: Springer. p. 190. doi:10.1007/978-1-84800-048-3. ISBN 978-1-84800-047-6.
  2. ^ Klenke, Achim (2008). Probability Theory. Berlin: Springer. p. 527. doi:10.1007/978-1-84800-048-3. ISBN 978-1-84800-047-6.
  3. ^ Kallenberg, Olav (2017). Random Measures, Theory and Applications. Switzerland: Springer. p. 87. doi:10.1007/978-3-319-41598-7. ISBN 978-3-319-41596-3.