Conditionally Positive Definite Kernels and Infinitely Divisible Distributions

We give a precise characterization of two important classes of conditionally positive definite (CPD) kernels in terms of integral transforms of infinitely divisible distributions. In particular, we show that for any stationary CPD kernel A(x, y) = f(x − y), f is the log-characteristic function of a uniquely determined infinitely divisible distribution; further, for any additive CPD kernel A(x, y) = g(x+ y), g is the log-moment generating function of a uniquely determined infinitely divisible distribution. The results strengthen the connections between CPD kernels and infinitely divisible distributions.