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 ∞ which belong to the second Wiener chaos (i.e. F ∞ 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.

A monotone scheme for nonlinear partial integro-differential equations with the convergence rate of $α$-stable limit theorem under sublinear expectation

“…To sum up, we conclude that q) , where Γ(α, q) = min{ 1 4 , 2−α 2α , q 2 }. This leads to the desired result.…”

“…On the other hand, if Θ is a singleton, (X t ) t≥0 is a classical Lévy process with triplet Θ, and X 1 is an α-stable random variable. The convergence rate of the classical α-stable central limit theorem has been studied in the Kolmogorov distance (see, e.g., [13, 15-17, 21, 24]) and in the Wasserstein-1 distance or the smooth Wasserstein distance (see, e.g., [1,10,11,23,30,38]). The first type is proved by the characteristic functions that do not exist in the sublinear framework, while the second type relies on Stein's method, which fails under the sublinear setting.…”

Bounding Kolmogorov distances through Wasserstein and related integral probability metrics

“…The classes H K , H W and H bW induce the Kolmogorov, Wasserstein (also known as the Kantorovich-Rubenstein or earth-mover's distance) and bounded Wasserstein distances (also known as the Fortet-Mourier or Dudley metric), which we denote by d K , d W and d bW , respectively. The classes H [m] and H m induce smooth Wasserstein distances, which we denote by d [m] and d m respectively (see, for example, [18] and [2]). Note that d [1] = d W and d 1 = d bW , and that d m ≤ d [m] for all m ≥ 1.…”

“…The case that Y is a Gaussian process has also recently been considered by [11]. Upper bounds on the Wasserstein distance d W = d [1] in terms of the d [2] and d [3] metrics have also been given in [18,31,47], and a bound on the total variation distance in terms of the Wasserstein metric when X and Y are random variables belonging to a finite sum of Wiener chaoses is given by [49].…”

Kolmogorov bounds for decomposable random variables and subgraph counting by the Stein-Tikhomirov method

“…The combination of elements of Stein's method with the theory of characteristic functions is sometimes called Stein-Tikhomirov method. Arras, Mijoule, Poly, and Swan [1] successfully used the Stein-Tikhomirov method to bound the convergence rate in contexts with non-Gaussian targets. Röllin [14] used the Stein-Tikhomirov method to bound the convergence rate in the Kolmogorov distance for normal approximation of normalized triangle counts in the Erdös-Rényi random graph.…”