Stationary Process

Stochastic process that is invariant to time shift.

Formula

Joint distribution of the stochastic process $\{X_i\}​$ is invariant to shifts in time index. $$ \begin{aligned} P[(X_1 = x_1), \ldots, (X_n = x_n)] &= P[(X_{1+k} = x_1), \ldots, (X_{n+k} = x_n)] \\ \forall n, k, \forall x^n &\in \mathcal{X}^n \end{aligned} $$

Theorem

If $\{X_i\}$ is a stationary process: $$ \begin{aligned} H(\mathcal{X}) = H'(\mathcal{X}) \end{aligned} $$

by Jon