stationary increment

A stochastic process $\{X(t)\mid t\in T\}$ of real-valuedrandom variables $X(t)$, where $T$ is a subset of $ℝ$, issaid have stationary increments if the probabilitydistribution function for $X(s+t)-X(s)$ is fixed (the same) for all$s\in T$ such that $s+t\in T$. In other words, the distribution for $X(s+t)-X(s)$ is a function of “how long” or $t$, not “when” or $s$.

A stochastic process that possesses both stationary increments andindependent increments is said to have stationary independentincrements.