Let $X$ be a set, and let $d:X\times X\rightarrow \mathbb{R}$ be a semimetric on that set (i.e. $\forall x,y\in X$, $d(x,y)=d(y,x)\ge 0$, and $d(x,y)=0$ iff $x=y$).
I seek conditions on $X$ and $d$ which are sufficient (and ideally necessary) for the following proposition:
- $\forall n\in \mathbb{N}$, $\forall x_1,x_2,\dots,x_n \in X$, the matrix $M=(m_{ij})_{i,j=1,\dots,n}$ with elements $m_{ij}=\exp{[-d(x_i,x_j)]}$ is positive definite.
Possible avenues of attack:
- When $X=\mathbb{R}$, and $d$ is the usual Euclidean metric, then these matrices are the covariance matrices of samples from standard Ornstein-Uhlenbeck processes, and hence positive definite.
- When $X=\mathbb{R}^n$, and $d$ is again the usual Euclidean metric, then, if I understand both result 504 here, and Bochner's theorem correctly, then again $M$ is guaranteed to be positive definite. (Honestly, I found the previously linked page rather incomprehensible, but the simple summary on the Wikipedia page for positive-definite functions gave me hope.)
- A direct approach via induction on $n$ may be possible, perhaps using the Schur-complement conditions for positive definite functions, and the fact that in the univariate case, the $M$ matrices have a tri-diagonal inverse.
The Bochner's theorem approach suggests we need $X$ to be a locally compact abelian group, and $d$ to come from a norm on $X$. Is this actually sufficient? Are such strong assumptions necessary? It's not even immediately clear to me that the triangle inequality should be necessary.
Thanks in advance for any help or suggestions.