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If you'll look up for the definition of a (discrete) white noise process on the web, you'll find some sources that say:

"white noise is defined by zero mean, finite and constant variance, zero autocorrelation"

and roughly the same amount of them that say:

"white noise is defined by zero mean, finite variance, zero autocorrelation".

In the latter (e.g. the Wikipedia page), "white noise" encompasses also the case of heteroskedastic white noise - where the variance isn't constant. Obviously, stationarity is dropped.

Which of these two is "the" standard definition? Am I missing something?

Note: I submitted the same question in math.stackexchange. Whenever I get an answer, I'll delete the other one.

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2 Answers 2

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There is no common definition of white noise.

You have two definitions. And there exists more definitions. For example, white noise is a stationary stochastic process with constant spectral density (and thus infinite variance).

https://www.encyclopediaofmath.org/index.php/White_noise

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    $\begingroup$ I agree with your first phrase! Finiteness of variance was the subject of a previous question, stats.stackexchange.com/questions/278263/…. $\endgroup$
    – Lo Scrondo
    Commented May 30, 2017 at 11:18
  • $\begingroup$ constant spectral density doesn't quite imply infinite variance. However, infinite variance is indeed possible with Cauchy white noise. $\endgroup$
    – carlo
    Commented Oct 14 at 19:05
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A white noise process has constant variance. So the second definition is incomplete.

Ref: Armitage encyclopedia of Biostatistics. The def there for discrete time requires that the autocorrelation looks like: $\gamma_{r,s} = \sigma \delta_{r,s}$.

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    $\begingroup$ Unfortunately, there's plenty of (qualified) sources that omit the constancy of variance. Someone even calls weak white noise the heteroskedastic one...My opinion is that there's no consensus on the definition of "white noise", and that there's not a theorical motivation behind it🙁. $\endgroup$
    – Lo Scrondo
    Commented May 30, 2017 at 11:49

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