Covariance Functions

Are all covariance functions known to be positive-definite?

Before we answer that question, let's define the meaning of positive-definite.

Positive-definiteness
Bochner Theorem (1955) states that a continuous function is positive definite if and only if it is the Fourier transform of a finite, non-negative measure.

And in linear algebra, a positive-definite matrix is a Hermitian matrix such that in many ways this matrix is analogous to a positive real number.

Let M be an n × n Hermitian matrix. The transpose of a vector, a, is denoted by a^T, and its conjugate transpose denoted by a^*. Matrix M is positive definite if and only if it satisfies the following property:

For all non-zero vectors z ∈ Cⁿ,

z^* M z > 0