[[Stochastic Processes|Stochastic process]] whose statistical properties (mean, [[Covariance and Variance|variance]], ...) does not change over time. >[!brainwaves] Intuition >Compare to stationary [[Probability Distribution|distributions]] for [[Markov Chain and Kernel|Markov chains]]. --- >[!info] Definition - Stationary Process >A stochastic process $\{X_t\}$ with unconditional (arbitrary starting value) [[Cumulative Distribution Function|cdf]] $F_X(x_{t_1+\tau},...x_{t_n+\tau})$ is stationary, if $F_X(x_{t_1+\tau},...x_{t_n+\tau})=F_X(x_{t_1},...x_{t_n}),\quad \forall \tau, t_1,t_n.$ **Examples** - [[Ornstein-Uhlenbeck Process]] admits stationary distribution, [[Wiener Process or Brownian Motion]] not