Chaos for rescaled measures on Kac’s sphere

Abstract

In this article we study a relatively novel way of constructing chaotic sequences of probability measures supported on Kac’s sphere, which are obtained as the law of a vector of N i.i.d. variables after it is rescaled to have unit average energy. We show that, as N increases, this sequence is chaotic in the sense of Kac, with respect to the Wasserstein distance, in $L^1$, in the entropic sense, and in the Fisher information sense. For many of these results, we provide explicit rates of polynomial order in N. In the process, we improve a quantitative entropic chaos result of Haurey and Mischler by relaxing the finite moment requirement on the densities from order 6 to $4+\mathit{ϵ}$.