A new approach to the Stein-Tikhomirov method: with applications to the second Wiener chaos and Dickman convergence

Abstract

In this paper, we propose a general means of estimating the rate at which convergences in law occur. Our approach, which is an extension of the classical Stein-Tikhomirov method, rests on a new pair of linear operators acting on characteristic functions. In principle, this method is admissible for any approximating sequence and any target, although obviously the conjunction of several favorable factors is necessary in order for the resulting bounds to be of interest. As we briefly discuss, our approach is particularly promising whenever some version of Stein's method applies. We apply our approach to two examples. The first application concerns convergence in law towards targets $F_\infty$ which belong to the second Wiener chaos (i.e. $F_{\infty}$ is a linear combination of independent centered chi-squared rvs). We detail an application to $U$-statistics. The second application concerns convergence towards targets belonging to the generalized Dickman family of distributions. We detail an application to a theorem from number theory. In both cases our method produces bounds of the correct order (up to a logarithmic loss) in terms of quantities which occur naturally in Stein's method.