Schoenberg (Ann. of Math. 39 (1938), 811-841) observed that if $f(t)$ is a completely monotonic function, then the radial kernel $K(x,y)=f(\|x-y\|^2)$ is positive definite on any Hilbert space. Since $$ f(t):=\frac{1}{1+\frac{t}{\sigma^2}} $$ is completely monotonic, the result follows.
GH from MO
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