Convergence in total variation to a mixture of Gaussian laws

It is not unusual that $X_noverset{dist}longrightarrow VZ$ where $X_n$, $V$, $Z$ are real random variables, $V$ is independent of $Z$ and $Zsimmathcal{N}(0,1)$. An intriguing feature is that $Pigl(VZin Aigr)=intmathcal{N}(0,V^2)(A),dP$ for each Borel set $Asubsetmathbb{R}$, 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 $d_{TV}(X_n,,VZ) ightarrow 0$ are given, where $d_{TV}(X_n,,VZ)$ is the total variation distance between the probability distributions of $X_n$ and $VZ$. 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; (ii) $X_n$ is related to the weighted quadratic variations of two independent Brownian motions.