Why a Random Walk Is Not Ergodic

Summary

The document asks whether a random walk, expressed as an autoregressive process with coefficient equal to one, is ergodic. The response says it is not: the process lacks the mean-reverting or centralizing behavior associated with a stationary AR model, and the usual law of large numbers argument for ergodicity does not apply. It contrasts this case with an AR process whose coefficient has absolute value below one.

The exchange offers a concise conclusion rather than a derivation. It does not define the precise form of ergodicity being used, develop formal conditions, or discuss variations such as random walks with drift or different innovation assumptions. Readers should therefore treat it as an introductory distinction between a unit-root process and a stable AR process, not as a full proof or a comprehensive account of ergodic theory. The concept matters when deciding whether long-run sample averages can stand in for process-level expectations.

Key ideas

• A random walk corresponds to an AR process with coefficient equal to one.
• The response characterizes the random walk as non-ergodic because it lacks a centralizing effect.
• The stated comparison is that an AR process is ergodic when the absolute value of its coefficient is below one.
• The discussion gives a short conceptual answer without formal proof or detailed assumptions.

Tags

# Ergodicity of Random Walk


# 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 ?

## Answer by userid is i (score 1)

https://quant.stackexchange.com/a/39816

A random walk is not ergodic, because there is no law of large numbers. The AR model is only ergodic if |phi|<1 to give a centralizing effect.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.