Convergence in Total Variation to a Mixture of Gaussian Laws

Abstract

It is not unusual that Xn⟶distVZ where Xn, V, Z are real random variables, V is independent of Z and Z∼N(0,1). An intriguing feature is that PVZ∈A=EN(0,V2)(A) for each Borel set A⊂R, 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.

Keywords

• mixture of Gaussian laws
• rate of convergence
• total variation distance
• Wasserstein distance
• weighted quadratic variation