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In a variety of contributions (e.g. Wikipedia) it seems that an infinite variance is linked (as a theoretical limit) only with continuos-time white noise processes.

However, in some papers/notes an infinite variance is considered also in the setting of discrete-time processes, and actually I could not see why, at least thereorically, it couldn't be so.

I think it's just a matter of conventions, but maybe I'm wrong and there's an univocal interpretation I'm missing...Someone could help?

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    $\begingroup$ Please provide some reference/link where a "white noise" process in discrete time is defined as having infinite variance. $\endgroup$ – Alecos Papadopoulos May 8 '17 at 13:57
  • $\begingroup$ E.g. people.stat.sfu.ca/~lockhart/richard/804/06_1/lectures/…: in the first slide, it is said that strong stationarity requires finite variance - which implies that in general it doesn't need to be finite. $\endgroup$ – Lo Scrondo May 8 '17 at 14:12
  • $\begingroup$ I cannot find any statement in your reference that says that. In general, stationary processes need not have finite variance. A "strong sense white noise," as defined on the first page, does have finite variance. $\endgroup$ – whuber May 8 '17 at 14:56
  • $\begingroup$ It's said indirectly @whuber. However, the last two phrases of your comment are what I was searching for. Indeed, do you agree with the answer offered by Alecos Papadopoulos? $\endgroup$ – Lo Scrondo May 8 '17 at 15:02
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    $\begingroup$ I cannot see where this was even said indirectly. Regardless, explicit definitions have to override any indirect inferences you might want to draw. One thing to beware of is that some authors might use technical terms in slightly different ways. That implies you must rely heavily on their definitions, read them literally, and bear those in mind when comparing different sources. $\endgroup$ – whuber May 8 '17 at 15:05
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The defining properties of a white-noise process in discrete time are: zero mean, zero autocorrelation, finite and constant variance.

The issues with continuous time white noise processes have to do with the peculiarities of the continuum.

But in discrete-time, we call a process white-noise only if it has the above three characteristics.

Strong-sense white noise, strengthens uncorrelatedness to full-stochastic independence, and equality of the first two moments to equality of distributions.

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  • $\begingroup$ Thank you @Alecos Papadopoulos. I'd just add that for some authors the concept of weak white noise contains also the case of heteroskedastic white noise, whose variance isn't constant. $\endgroup$ – Lo Scrondo May 9 '17 at 4:36

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