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 <math> (X_t)_{t \in T} </math> be a stochastic process. In most cases, <math> T= \N </math> or <math> T=\R^+ </math>. Then the stochastic process has independent increments if and only if for every <math> m \in \N </math> and any choice <math> t_0, t_1, t_2, \dots,t_{m-1}, t_m \in T</math> with

:<math> t_0 < t_1 < t_2< \dots < t_m </math>

the random variables

:<math> (X_{t_1}-X_{t_0}),(X_{t_2}-X_{t_1}), \dots, (X_{t_m}-X_{t_{m-1 )</math>

are stochastically independent.

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