Independent increments

A stochastic process has independent increments when, for every $t_0<t_1<\cdots <t_k$, the random variables $X_{t_1}-X_{t_0},\dots ,X_{t_k}-X_{t_{k-1}}$ 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, $\mathrm{Cov}(S_m,S_n)=\min (m,n)$ 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.