Convergence in total variation to a mixture of Gaussian laws
type: Item
instance of: scholarly article
title: Convergence in total variation to a mixture of Gaussian laws
zbMATH Open document ID: 1404.60008
author: Luca Pratelli
author: Pietro Rigo
published in: Mathematics
publication date: 2018-12-18T00:00:00Z
review text: Summary: It is not unusual that \(X_n \mathop{\operatorname{dist}}\limits_{\longrightarrow} V Z\) where \(X_n\), \(V\), \(Z\) are real random variables, \(V\) is independent of \(Z\) and \(Z \sim \mathcal N(0, 1)\). An intriguing feature is that \(P(VZ\in A) = E \left\{\mathcal N,(0,V^2)(A)\right\}\) for each Borel set \(A \subset \mathbb R\), namely, the probability distribution of the limit \(VZ\) is a mixture of centered Gaussian laws with (random) variance \(V^2\). In this paper, conditions for \(d_{TV}(X_n, VZ) \rightarrow 0\) are given, where \(d_{TV}(X_n, VZ)\) is the total variation distance between the probability distributions of \(X_n\) and \(V Z\). To estimate the rate of convergence, a few upper bounds for \(d_{TV}(X_n, VZ)\) are given as well. Special attention is paid to the following two cases: (i) \(X_n\) is a linear combination of the squares of Gaussian random variables; and (ii) \(X_n\) is related to the weighted quadratic variations of two independent Brownian motions.
Mathematics Subject Classification ID: 60B10
Mathematics Subject Classification ID: 60F05
zbMATH DE Number: 6994630
zbMATH Keywords: mixture of Gaussian laws
zbMATH Keywords: rate of convergence
zbMATH Keywords: total variation distance
zbMATH Keywords: Wasserstein distance
zbMATH Keywords: weighted quadratic variation
MaRDI profile type: Publication
full work available at URL: math6060099
OpenAlex ID: W2807494317
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DOI: 10.3390/MATH6060099
description: scientific article; zbMATH DE number 6994630