Mercer's Theorem

The stochastic analog of Mercer's theorem is the Karhunen-Loève theorem, which decomposes a random process into orthogonal eigenfunctions with random coefficients. Where Mercer guarantees that a deterministic kernel has a spectral decomposition, Karhunen-Loève guarantees that a stochastic process with continuous covariance has a similar decomposition, making it the foundation of principal component analysis in function spaces.

See also: Karhunen-Loève Theorem