Let $K(s,t)$ be a real function over $T\times T$, where $T$ is arbitrary. $K$ has two properties:
- $K$ is symmetric ($K(s,t)=K(t,s)$).
- $K$ is nonnegative-definite ($\sum_{i,j=1}^k K(t_i,t_j)x_ix_j\geq 0$, for $k\geq 1$, $t_1,\ldots,t_k \in T$ and $x_1,\ldots,x_k$ real).
I need to show that there exist a process $[X_t:t\in T]$ for which $(X_{t_1},\ldots,X_{t_k})$ has the centered normal distribution with covariances $K(t_i,t_j),~ i,j=1,\ldots,k$.
Any help would be greatly appreciated.