An extension of Kolmogorov’s theorem for continuous covariances

Abstract:

The theorem of Kolmogorov stating that a non-negative definite kernel on ${N^1} \times {N^{ - 1}}$ is the covariance of a stochastic process on ${N^1}$ is generalized to continuous nonnegative definite functions on $Y \times Y,Y$ being a separable Hausdorff space. Also, a representation of such continuous nonnegative definite functions and their associated stochastic processes is provided.