It is proved that a continuous function f on ℝ n is negative definite if and only if it is polynomially bounded and satisfies the inequality for all i.i.d. random vectors X and Y in ℝ n . The proof uses Fourier transforms of tempered distributions. The "only if" part has been proved earlier by Lifshits et al. (A probabilistic inequality related to negative definite functions.
Keywords: 42A82; 60E10; Fourier inversion theorem; Lévy–Khintchine representation; Negative definite function; Polynomially bounded; Primary: 60E15; Secondary: 42B10.