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

$\mathrm {Cov} (x,y)=\int _{H}\langle x,z\rangle \langle y,z\rangle \,\mathrm {d} \mathbf {P} (z)$

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

$\mathrm {Cov} (x,y)=\langle Cx,y\rangle$

(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.

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

$\mathrm {Cov} (x,y)=\int _{B}\langle x,z\rangle \langle y,z\rangle \,\mathrm {d} \mathbf {P} (z)$

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

$\mathrm {Cov} (x,y)=\int z(x)z(y)\,\mathrm {d} \mathbf {P} (z)=E(z(x)z(y))$

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.

See also

• Abstract Wiener space – Mathematical construction relating to infinite-dimensional spaces.
• Cameron–Martin theorem – Theorem defining translation of Gaussian measures (Wiener measures) on Hilbert spaces.
• Feldman–Hájek theorem – Theory in probability theory
• Structure theorem for Gaussian measures – Mathematical theorem

Further reading

• Baker, C. R. (September 1970). On Covariance Operators. Mimeo Series. Vol. 712. University of North Carolina at Chapel Hill.
• Baker, C. R. (December 1973). "Joint Measures and Cross-Covariance Operators" (PDF). Transactions of the American Mathematical Society. 186: 273–289.
• Vakhania, N. N.; Tarieladze, V. I.; Chobanyan, S. A. (1987). "Covariance Operators". Probability Distributions on Banach Spaces. Dordrecht: Springer Netherlands. pp. 144–183. doi:10.1007/978-94-009-3873-1_3. ISBN 978-94-010-8222-8. Retrieved 2024-04-11.