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## Definition for stochastic processes

## Definition for random measures

## Independent S-increments

## Application

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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, all additive process^{}
and the Poisson point process.

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.^{}

A random measure has got independent increments if and only if the random variables are stochastically independent for every selection of pairwise disjoint measurable sets and every . ^{}

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.^{}

Independent increments are a primaryproperty 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 necessaryrole in the characterization of Poisson point process and unlimiteddivisibility

- Sato, Ken-Ito (1999).
*Lévy processes and infinitely divisible distributions*. Cambridge University Press. pp. 31–68. ISBN 9780521553025. - Klenke, Achim (2008).
*Probability Theory*. Berlin: Springer. p. 190. doi:. ISBN 978-1-84800-047-6. - Klenke, Achim (2008).
*Probability Theory*. Berlin: Springer. p. 527. doi:. ISBN 978-1-84800-047-6. - Kallenberg, Olav (2017).
*Random Measures, Theory and App*. Switzerland: Springer. p. 87. doi:. ISBN 978-3-319-41596-3.

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