Covariance operator

Template:Short description In probability theory, for a probability measure P on a Hilbert space H with inner product $⟨ \cdot, \cdot ⟩$ , the covariance of P is the bilinear form Cov: H × H → R given by

C o v (x, y) = \int_{H} ⟨ x, z ⟩ ⟨ y, z ⟩ d 𝐏 (z)

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

C o v (x, y) = ⟨ C x, y ⟩

(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

C o v (x, y) = \int_{B} ⟨ x, z ⟩ ⟨ y, z ⟩ d 𝐏 (z)

where $⟨ x, z ⟩$ 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

C o v (x, y) = \int z (x) z (y) d 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.

References