Stability of the Markov-Krein Transform Under Classical Convolutions

Abstract

We prove that, for every $c>0$, the image of the generalized Markov-Krein transform on probability measures with finite logarithmic moment is closed under classical additive convolution and under the Mellin convolution. This resolves the positivity conjecture for the additive high-temperature convolution, a one-parameter interpolation between classical and free convolution arising in random matrix theory and representation theory in the regime $N\to\infty, Nβ_N/2\to c$. Our proof relies on a convolution theorem for universally prestarlike functions.