General Upper Bounds by Fourier Methods

Abstract

The Fourier methodology developed in [91] to study the Stein’s equation in the Gaussian setting, often nowadays referred to as the Stein–Tikhomirov method, has been extended in [4] to provide rates of convergence in Kolmogorov or in smooth Wasserstein distance for sequences $(X_n)_{n\ge 1}$ converging toward $X_\infty$. This approach leads to quantitative estimates when $X_\infty$ is a second-order Wiener chaos, or the generalized Dickman distribution or even the symmetric $\alpha$-stable one. Corollary 3.5, or Proposition 3.8, or even the stable characterizing identities of the previous chapter allow extensions of the aforementioned estimates to classes of infinitely divisible sequences. The forthcoming results are general and have a non-empty intersection with those on the Dickman distribution presented in [4].