In probability theory, for a probability measure P on a Hilbert space H with inner product ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle } , the covariance of P is the bilinear form Cov: H × H → R given by
C o v ( x , y ) = ∫ H ⟨ x , z ⟩ ⟨ y , z ⟩ d P ( z ) {\displaystyle \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
C o v ( x , y ) = ⟨ C x , y ⟩ {\displaystyle \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
C o v ( x , y ) = ∫ B ⟨ x , z ⟩ ⟨ y , z ⟩ d P ( z ) {\displaystyle \mathrm {Cov} (x,y)=\int _{B}\langle x,z\rangle \langle y,z\rangle \,\mathrm {d} \mathbf {P} (z)}
where ⟨ x , z ⟩ {\displaystyle \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
C o v ( x , y ) = ∫ z ( x ) z ( y ) d P ( z ) = E ( z ( x ) z ( y ) ) {\displaystyle \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 ↦ u ( x ) {\displaystyle u\mapsto u(x)} evaluated at z.
See also Abstract Wiener space – Mathematical construction relating to infinite-dimensional spaces Cameron–Martin theorem – Theorem describing translation of Gaussian 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. doi:10.1090/S0002-9947-1973-0336795-3. 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.
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