Convergence in Law Implies Convergence in Total Variation for Polynomials in Independent Gaussian, Gamma or Beta Random Variables

Abstract

Consider a sequence of polynomials of bounded degree evaluated in independent Gaussian, Gamma or Beta random variables. We show that, if this sequence converges in law to a nonconstant distribution, then (i) the limit distribution is necessarily absolutely continuous with respect to the Lebesgue measure and (ii) the convergence automatically takes place in the total variation topology. Our proof, which relies on the Carbery-Wright inequality and makes use of a diffusive Markov operator approach, extends the results of [9] to the Gamma and Beta cases.

Fractional smoothness of images of logarithmically concave measures under polynomials

“…provided that σ f , σ g ≥ a. This estimate generalizes some recent results from [25,26] and [11] to the case of log-concave measures. However, even in the case of a Gaussian measure the power at the Fortet-Mourier distance in our estimate is better in comparison with similar results from the cited papers.…”

“…Proof. We apply Lemma 2.1 and Lemma 2 Our next corollary generalizes some results from [25,26] and [11] to the case of arbitrary logconcave measures in place of Gaussian measures, and even in the Gaussian case this estimate provides a better rate of convergence as compared to analogous estimates from [25] and [11] in the one-dimensional case. It follows directly from Lemma 2.3 and Theorem 5.1.…”

“…This class of distributions ν is of interest for many applications, because it contains typical statistics and the class of all polynomials of a fixed degree can be considered as an important family of nonlinear transformations of a given measure. Various properties of measures in the class under consideration have been studied in many works, see [11,12,13,17,24,25,27,28] for the case of Gaussian measures and [1,6,7,15,22,26] for the case of general logarithmically concave measures.…”

Weak Convergence and Weak Convergence

“…All nondegenerate random variables in GGC have probability density functions and they are unimodel, see part vi) of [2] for this (also see the introduction of [18]). Then the claim follows from [15] (also see page 383 of [13]). Since the limit distribution Z has probability density function convergence in law implies convergece in the Kolmogorov distance, a fact that can be derived by using Dini's second theorem (see the introduction of [13] for this).…”

“…Then the claim follows from [15] (also see page 383 of [13]). Since the limit distribution Z has probability density function convergence in law implies convergece in the Kolmogorov distance, a fact that can be derived by using Dini's second theorem (see the introduction of [13] for this).…”