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

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