Covariance Operator

In probability theory, for a probability measure P on a Hilbert space ''H'' with inner product $\langle \cdot,\cdot\rangle$, the covariance of P is the bilinear form Cov: ''H'' × ''H'' → R given by

:$\mathrm(x, y) = \int_ \langle x, z \rangle \langle y, z \rangle \, \mathrm \mathbf (z)$

for all ''x'' and ''y'' in ''H''. The covariance operator ''C'' is then defined by

:$\mathrm(x, y) = \langle Cx, y \rangle$

(from the Riesz representation theorem, such operator exists if Cov is bounded). Since Cov is symmetric in its arguments, the covariance operator is
self-adjoint. When P is a centred Gaussian measure, ''C'' is also a nuclear operator. In particular, it is a compact operator of trace class, that is, it has finite trace.

Even more generally, for a probability measure P on a Banach space ''B'', the covariance of P is the bilinear form on the algebraic dual ''B''^#, defined by

:$\mathrm(x, y) = \int_ \langle x, z \rangle \langle y, z \rangle \, \mathrm \mathbf (z)$

where $\langle x, z \rangle$ is now the value of the linear functional ''x'' on the element ''z''.

Quite similarly, the covariance function of a function-valued random element (in special cases is called random process or random field) ''z'' is

:$\mathrm(x, y) = \int z(x) z(y) \, \mathrm \mathbf (z) = E(z(x) z(y))$

where ''z''(''x'') is now the value of the function ''z'' at the point ''x'', i.e., the value of the linear functional $u \mapsto u(x)$ evaluated at ''z''.

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Probability theory
Covariance and correlation
Bilinear forms