White noise ($\mathrm{WN}(0,{\sigma }^2)$)

For the AR(1) and MA(1) formulas here, white noise means a real sequence indexed by all integer times withThese 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 ${\epsilon }_t=A{Z}_t$. Its mean is zero, its variance is $\mathbb{E}[A^2]=5/2$, and its nonzero-lag autocovariances vanish. Nevertheless every $|{\epsilon }_t|$ 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.

Discrete Gaussian white noise is a jointly Gaussian sequence with expected value zero and covariance ${\sigma }^2$ at lag zero and zero at other lags. Joint normality makes the components independent. This is a sequence of ordinary random variables, unlike the generalized continuous-time white-noise process.

Continuous-time Gaussian white noise is a random distribution rather than an ordinary finite-variance function at every time. Integrating it against deterministic kernels gives Gaussian random variables with covariance $q\int f(s)g(s)ds$. For independent isotropic components, a three-dimensional dot covariance has amplitude $3q$. An ordinary rapidly decaying force covariance approximates this model only above its correlation time.

A sequence of independent and identically distributed random variables with zero mean and finite variance is strong white noise. It is weak white noise, but need not have a normal distribution.

A white noise sequence has zero mean, a common finite variance and zero autocovariance at every nonzero lag. This requires uncorrelated random variables, rather than independent random variables.