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Mathematics 2018, 6(6), 99; https://doi.org/10.3390/math6060099

Convergence in Total Variation to a Mixture of Gaussian Laws

1
Accademia Navale, Viale Italia 72, 57100 Livorno, Italy
2
Dipartimento di Matematica “F. Casorati”, Universita’ di Pavia, via Ferrata 1, 27100 Pavia, Italy
*
Author to whom correspondence should be addressed.
Received: 29 April 2018 / Revised: 1 June 2018 / Accepted: 5 June 2018 / Published: 11 June 2018
(This article belongs to the Special Issue Stochastic Processes with Applications)
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Abstract

It is not unusual that XndistVZ where Xn, V, Z are real random variables, V is independent of Z and ZN(0,1). An intriguing feature is that PVZA=EN(0,V2)(A) for each Borel set AR, namely, the probability distribution of the limit VZ is a mixture of centered Gaussian laws with (random) variance V2. In this paper, conditions for dTV(Xn,VZ)0 are given, where dTV(Xn,VZ) is the total variation distance between the probability distributions of Xn and VZ. To estimate the rate of convergence, a few upper bounds for dTV(Xn,VZ) are given as well. Special attention is paid to the following two cases: (i) Xn is a linear combination of the squares of Gaussian random variables; and (ii) Xn is related to the weighted quadratic variations of two independent Brownian motions. View Full-Text
Keywords: mixture of Gaussian laws; rate of convergence; total variation distance; Wasserstein distance; weighted quadratic variation mixture of Gaussian laws; rate of convergence; total variation distance; Wasserstein distance; weighted quadratic variation
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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Pratelli, L.; Rigo, P. Convergence in Total Variation to a Mixture of Gaussian Laws. Mathematics 2018, 6, 99.

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