Approximation properties of Gamma operators

AbstractIn this paper the approximation properties of Gamma operators Gn are studied to the locally bounded functions and the absolutely continuous functions, respectively. Firstly, in Section 2 of the paper a quantitative form of the central limit theorem in probability theory is used to derive an asymptotic formula on approximation of Gamma operators Gn for sign function. And then, this asymptotic formula combining with a metric form Ωx(f,λ) is used to derive an asymptotic estimate on the rate of convergence of Gamma operators Gn for the locally bounded functions. Next, in Section 3 of the paper the optimal estimate on the first order absolute moment of the Gamma operators Gn(|t−x|,x) is obtained by direct computations. And then, this estimate and Bojanic–Khan–Cheng's method combining with analysis techniques are used to derive an asymptotically optimal estimate on the rate of convergence of Gamma operators Gn for the absolutely continuous functions

Stein's method for comparison of univariate distributions

We propose a new general version of Stein's method for univariate distributions. In particular we propose a canonical definition of the Stein operator of a probability distribution {which is based on a linear difference or differential-type operator}. The resulting Stein identity highlights the unifying theme behind the literature on Stein's method (both for continuous and discrete distributions). Viewing the Stein operator as an operator acting on pairs of functions, we provide an extensive toolkit for distributional comparisons. Several abstract approximation theorems are provided. Our approach is illustrated for comparison of several pairs of distributions : normal vs normal, sums of independent Rademacher vs normal, normal vs Student, and maximum of random variables vs exponential, Frechet and Gumbel.Comment: 41 page

Stein's method on the second Wiener chaos : 2-Wasserstein distance

In the first part of the paper we use a new Fourier technique to obtain a Stein characterizations for random variables in the second Wiener chaos. We provide the connection between this result and similar conclusions that can be derived using Malliavin calculus. We also introduce a new form of discrepancy which we use, in the second part of the paper, to provide bounds on the 2-Wasserstein distance between linear combinations of independent centered random variables. Our method of proof is entirely original. In particular it does not rely on estimation of bounds on solutions of the so-called Stein equations at the heart of Stein's method. We provide several applications, and discuss comparison with recent similar results on the same topic

Convergence of a Boundary Integral Method for Water Waves

We prove nonlinear stability and convergence of certain boundary integral methods for time-dependent water waves in a two-dimensional, inviscid, irrotational, incompressible fluid, with or without surface tension. The methods are convergent as long as the underlying solution remains fairly regular (and a sign condition holds in the case without surface tension). Thus, numerical instabilities are ruled out even in a fully nonlinear regime. The analysis is based on delicate energy estimates, following a framework previously developed in the continuous case [Beale, Hou, and Lowengrub, Comm. Pure Appl. Math., 46 (1993), pp. 1269–1301]. No analyticity assumption is made for the physical solution. Our study indicates that the numerical methods must satisfy certain compatibility conditions in order to be stable. Violation of these conditions will lead to numerical instabilities. A breaking wave is calculated as an illustration