Covariance operator

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

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

(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 (the infinite-dimensional analogy of the transposition symmetry in the finite-dimensional case). 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

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 called random process or random field) z is

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