Convergence in total variation to a mixture of Gaussian laws
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.
scientific article; zbMATH DE number 6994630
scientific article