Ergodicity of Random Walk

The Random walk is a special case of AR(1) with

$x_t = \phi x_{t-1} + \epsilon_t$ with $\phi = 1$

A process is ergodic if two samples of a stochasitc process sampled far (say j < $\infty$ ) apart are independent.

Can someone help me know if the random walk is ergodic and how ?

• Are you sure of your definition of ergodicity? A theorem known to me is "a stationary process is ergodic if any two random variables positioned far apart in the sequence are almost independently distributed". But this theorem cannot be applied to a random walk, since it is not stationary. The general definition of ergodicity is another one. – noob2 May 15 '18 at 0:44