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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#### Best Answer

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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