Convergence in total variation to a mixture of Gaussian laws

It is not unusual that $X_n\overset{dist}\longrightarrow VZ$ 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 $P\bigl(VZ\in A\bigr)=E\Bigl\{\mathcal{N}(0,V^2)(A)\Bigr\}$ 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 $VZ$. 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.