= White noise
{title2=$\operatorname{WN}(0,\sigma^2)$}
{wiki}
= White noise process
{synonym}
For the <AR(1)> and <MA(1)> formulas here, <white noise> means a real sequence indexed by all integer times with
$$
\mathbb E[\varepsilon_t]=0,\qquad
\operatorname{Cov}(\varepsilon_s,\varepsilon_t)=\sigma^2\mathbf1_{\{s=t\}},\qquad 0<\sigma^2<\infty.
$$
These are the <weak white noise> assumptions: zero <mean>, a common finite positive <variance>, and <uncorrelated random variables> at distinct times. Independence and a <normal distribution> are additional assumptions. <Strong white noise> requires <independent and identically distributed random variables>; <discrete Gaussian white noise> is the jointly <Gaussian> special case. Some definitions allow a constant nonzero <mean>; subtract it before using the centered formulas here.
For a concrete dependent example, let $A$ equal $1$ or $2$ with equal probabilities, independently of a sequence of <independent> <Rademacher random variables> $Z_t$, and put $\varepsilon_t=AZ_t$. Its <mean> is zero, its <variance> is $\mathbb E[A^2]=5/2$, and its nonzero-lag <autocovariances> vanish. Nevertheless every $|\varepsilon_t|$ equals the same $A$: seeing one magnitude determines all the others. This is <white noise> with dependence invisible to its <ACF>, and is also an example of the issue discussed in <ergodic stationary process>.
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