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.[2]
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 . [3]
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.[4]
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
Sato, Ken-Ito (1999). Lévy processes and infinitely divisible distributions. Cambridge University Press. pp. 31–68. ISBN 9780521553025.