Gaussian white noise

Gaussian white noise is a generalized Gaussian process with zero mean and covariance proportional to a Dirac delta function in time, and possibly in space. It is the formal time derivative of Brownian motion and has independent fluctuations on disjoint time intervals.

Observe $Y(t)={\int }_0^{t}f(s)ds+\sigma W(t)$ on $[0,1]$, where $f\in L^2[0,1]$ is an unknown deterministic drift and $W$ is standard Brownian motion. The observation against a deterministic test function is $Y(h)=⟨f,h⟩+\sigma W(h)$, with covariance ${\sigma }^2⟨h,g⟩$. Gaussian white noise is represented by an isonormal Gaussian process on test functions rather than a pointwise noise function. Taking $\sigma =n^{-1/2}$ is the usual statistical scaling.

Generalized Gaussian white noise over a real Hilbert space $H=L^2$ is an isonormal Gaussian process $W(h)$ with covariance $⟨h,g{⟩}_H$. Its law $P_0$ is carried on a larger observation space; the identity is not the covariance of an infinite-dimensional $H$-valued Gaussian measure because it is not a trace-class operator. A deterministic signal $h\in H$ defines a translated observation law $P_h$ with Radon-Nikodym derivativeIn an orthonormal basis, $W_m(h)={\sum }_j{m}_j{h}_j$ is a stochastic series with total variance $\Vert h{\Vert }_H^2$, not an inner product of two $H$-valued observations. The formula follows from the Cameron-Martin theorem for a Gaussian measure in its generalized white-noise version, or from finite-dimensional Gaussian likelihood ratios. On ${\mathbb{R}}^2$, generalized noise can be realized on tempered distributions; an unweighted global negative Sobolev space is not automatically a suitable almost-sure support on an unbounded domain. The stochastic series avoids imposing that unsupported regularity.

The Gaussian sequence model observeswhere the $g_k$ are independent standard normal variables and the unknown signal commonly lies in ${\ell }^2$. It is the coordinate representation of a Gaussian white-noise experiment.

A Bernstein-von Mises theorem says that a suitably centered and scaled posterior distribution converges to the normal distribution prescribed by the local likelihood. For a fixed linear functional in a Gaussian sequence model and a prior with locally flat positive density, the limiting posterior variance is the squared ${\ell }^2$ norm of the functional's coefficient vector.

A Gaussian net test covers a composite alternative by finitely many metric balls and takes the maximum of the likelihood-ratio tests against their centers. Gaussian tail bounds control each test, while a union bound costs the logarithm of the covering number.

Because Gaussian white noise is not an ${\ell }^2$ vector, least squares over $\Theta \subset {\ell }^2$ is defined by maximizingover $\theta \in \Theta$, with the noise pairing interpreted through an isonormal Gaussian process. Differences of this contrast equal the formal differences of squared residual norms.