On Bobkov's approximate de Finetti representation via approximation of permanents of complex rectangular matrices

TL;DR

This paper improves the approximation bounds for permanents of complex matrices related to symmetric probability measures, simplifying previous methods and enabling higher order accuracy in de Finetti representations.

Contribution

It introduces an asymptotic expansion approach to replace induction, providing new explicit higher order bounds for permanents and associated probability measures.

Findings

New explicit higher order approximation bounds for permanents

Simplified method avoiding induction in proofs

Enhanced accuracy in approximate de Finetti representations

Abstract

Bobkov (J. Theoret. Probab. 18(2) (2005) 399-412) investigated an approximate de Finetti representation for probability measures, on product measurable spaces, which are symmetric under permutations of coordinates. One of the main results of that paper was an explicit approximation bound for permanents of complex rectangular matrices, which was shown by a somewhat complicated induction argument. In this paper, we indicate how to avoid the induction argument using an (asymptotic) expansion. Our approach makes it possible to give new explicit higher order approximation bounds for such permanents and in turn for the probability measures mentioned above.