Ergodicity, Return Series, and the Planar Brownian Motion Example

Summary

The document asks whether financial return series are ergodic, whether a statistic can test that property, and whether conclusions depend on sampling frequency. It mentions inspecting the autocorrelation function for decay as an intuitive idea, but does not establish that this is a valid ergodicity test or provide a procedure for choosing a lag threshold.

The answer offers a mathematical example involving planar Brownian motion, described as an exceptional non-ergodic Brownian motion. It notes the unusual combination of repeatedly revisiting some points while leaving other regions unreachable. This is a conceptual pointer rather than an analysis of observed asset returns: no return data, test statistic, sampling-rate comparison, or formal proof is included. The example therefore motivates care when connecting time-series behavior to ergodicity, but does not settle whether a particular market series is ergodic.

Key ideas

• The question distinguishes ergodicity from the observed decay of autocorrelation.
• Autocorrelation decay is proposed as an intuition, not validated as a sufficient test.
• Planar Brownian motion is presented as an example of non-ergodic behavior.
• The discussion provides no empirical test for financial returns or sampling frequencies.

Tags

# Are return time series ergodic?


# Are return time series ergodic?












It seems intuitive to me that return time series would be ergodic. Is there a test statistic that I can use to check this? Would this be affected by sampling rate?

One way I can think of checking ergodicity is to plot the ACF and check if the autocorrelation decays with sufficiently large lag (which it will).

## Answer by Dave Harris (score 1)

https://quant.stackexchange.com/a/63807

There is a proof that may impinge on this answer. If returns are r(P1,P2) and (P1,P2) follow a Brownian motion, then it is proven that planar Brownian motion is the only Brownian motion that is not ergodic.

It has the interesting property that any point reached $(\alpha,\beta)$ will be returned to an infinite number of times, but also that there will exist many holes that will never be reached.

I don't have the book where I am at, but I believe the proof is in Brownian Motion by Peter Morters and Yuval Peres. ISBN-13: 978-0521760188

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