词条 | Independent increments |
释义 |
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 processesLet 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 measuresA 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-incrementsLet 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] ApplicationIndependent 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 1 : Probability theory |
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