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Is there a white noise which is not ergodic?

How is the ergodicty of a white noise tested?

Thanks!

Note: A white noise is defined as in Time Series: Theory and Methods By Peter J. Brockwell, Richard A. Davis:

Definition 3.1.1 . The process $\{ Z_t \}$ is said to be white noise with mean $0$ and variance $\sigma^2$ , written $\{Z_t\} \sim WN(0, \sigma^2 )$, if and only if $\{ Z_t \}$ has zero mean and covariance function $$\gamma(h) = \sigma^2, \text{ if }h=0; $$ $$\gamma(h) =0, \text{ if } h \neq 0.$$

I believe it is the most common definition of white noise. Also I don't see how it implies ergodicity.

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  • $\begingroup$ I think ergodicity comes in the definition of the white noise (whiteness is stronger than ergodicity). $\endgroup$ – Daniel Feb 15 '14 at 20:44
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    $\begingroup$ The answer depends on the definition of "white noise": by some definitions it is (trivially) ergodic whereas other broader definitions allow for it to be non-ergodic. What is your definition? $\endgroup$ – whuber Feb 15 '14 at 22:12
  • $\begingroup$ @whuber: See Time Series: Theory and Methods By Peter J. Brockwell, Richard A. Davis $\endgroup$ – Tim Feb 15 '14 at 23:52
  • $\begingroup$ @Daniel: can you tell me the reference of you definition, and what the definition is? I update my post with the definition I use. $\endgroup$ – Tim Feb 15 '14 at 23:55
  • $\begingroup$ The definition you posted must have a typographical error because it contradicts itself and is incomplete. You probably mean "$h\ne 0$" on the last line. $\endgroup$ – whuber Feb 16 '14 at 17:23
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I am not sure if it can be used for a time series, but there is a function is.matrix_ergodic {popdemo} which tests ergodicity of a matrix.

I also think that definition of white noise is stronger than that of ergodicity, because a process can be ergodic even if it is not independent, because it refers to asymptotic property. But independence is often formulated as third condition of whiteness, in addition to time independent first and second moments.

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    $\begingroup$ Which definition of white noise are you applying? According to some definitions, even independence of a "white noise" process does not imply ergodicity. $\endgroup$ – whuber Feb 15 '14 at 22:13
  • $\begingroup$ I am refering to the usual definition of white noise. Isn't it correct? $\endgroup$ – DatamineR Feb 15 '14 at 22:21
  • $\begingroup$ There are multiple (quite different) definitions of white noise even in the Wikipedia article on the subject. The key issue is whether the random variables $Z_t$ have identical distributions or not. If they do and they are independent, then (trivially) the process is ergodic: that follows immediately from the definitions. If they do not have identical distributions, then the process might not be ergodic. For instance, let $Z_t$ be independent and have a Normal$(0, \sigma^2/t)$ distribution for $t=1, 2, \ldots.$ That's why your answer needs clarification. $\endgroup$ – whuber Feb 16 '14 at 17:24
  • $\begingroup$ @whuber: At least in the definition I am using, WN must be stationary. In your example, the variance is not constant over time. $\endgroup$ – Tim Feb 16 '14 at 17:45
  • $\begingroup$ That's precisely why I originally asked you to state your definitions, Tim! $\endgroup$ – whuber Feb 16 '14 at 17:48
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Broadly speaking, a dynamical system is ergodic if it has the same behavior averaged over time as averaged over the space of all the system's states. Mean and covariance of white noise can be determined from its time average, hence it is ergodic. Simple noise models are usually ergodic; however, when the noise depends on the system, then its characteristics may change over time. Time average in such a case may not be equal to the ensemble average (hence not ergodic).

For example think of a system with two modes. At mode1 let the noise be white. At mode2 let the noise be poisson with parameter λ. Let the system be stuck with one of the modes depending on its initial condition. Then the time average of the noise would either be 0 or λ. You may argue that depending on the initial state probability (i.e. p vs. 1-p) the time average should be (1-p)λ, but that is not true (as you may see, time average definition does not include anything about the initial state probability so the best you can say is that the time average is either 0 or λ.) In fact, (1-p)λ is the ensemble average which is different from the time average by which we may conclude that the noise is not ergodic.

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