= Independent increments
A <stochastic process> has independent increments when, for every $t_0<t_1<\cdots<t_k$, the <random variables> $X_{t_1}-X_{t_0},\ldots,X_{t_k}-X_{t_{k-1}}$ are <independent>. Adjacent intervals are allowed: each <increment> represents the change after its left endpoint up to its right endpoint. A <Poisson process>, <Brownian motion>, and a <random walk> with independent steps have this property. Their values at different times need not be independent: for a fair <simple random walk on the integer line>, $\operatorname{Cov}(S_m,S_n)=\min(m,n)$ because both positions contain the same first $\min(m,n)$ increments. <Independent increments> need not be <stationary increments>; summing independent steps with time-dependent <variances> gives a counterexample.
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