4. General Upper Bounds by Fourier Methods

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].