Covariance operator

Summary

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.

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 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

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

See also

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Further reading

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  • 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.

References

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