Distances between probability distributions via characteristic functions and biasing

In a spirit close to classical Stein’s method, we introduce a new technique to derive first order ODEs on differences of characteristic functions. Then, using concentration inequalities and Fourier transform tools, we convert this information into sharp bounds for the so-called smooth Wasserstein metrics which frequently arise in Stein’s method theory. Our methodolgy is particularly efficient when the target density and the object of interest both satisfy biasing equations (including the zero-bias and size-bias mechanisms). In order to illustrate our technique we provide estimates for: (i) the Dickman approximation of the sum of positions of records in random permutations, (ii) quantitative limit theorems for convergence towards stable distributions, (iii) a general class of infinitely divisible distributions and (iv) distributions belonging to the second Wiener chaos, that is to say, in some situations impervious to the standard procedures of Stein’s method. Except for the stable distribution, our bounds are sharp up to logarithmic factors. We also describe a general argument and open the way for a wealth of refinements and other applications.