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, all additive process and the Poisson point process. Definition for stochastic processes. Let formula_1 be a stochastic process. In most cases, formula_2 or formula_3. Then the stochastic process has independent increments if and only if for every formula_4 and any choice formula_5 with the random variables are stochastically  independent. Definition for random measures. A random measure formula_8 has got independent increments if and only if the random variables formula_9 are stochastically independent for every selection of pairwise disjoint measurable sets formula_10 and every formula_4. Independent S-increments. Let formula_8 be a random measure on formula_13 and define for every bounded measurable set formula_14 the random measure formula_15 on formula_16 as Then formula_8 is called a random measure with independent S-increments, if for all bounded sets formula_19 and all formula_20 the random measures formula_21 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 Poisson point process and infinite divisibility