On conditions for convergence of the densities of smoothed distributions in the central limit theorem (Q1358029)

Let \(\xi\), \(\xi_1, \xi_2, \dots\) be a sequence of i.i.d. random variables with mean zero and variance 1. Set \(S_n= \xi_1+ \cdots +\xi_n\). Let \(\nu\) be a random variable having a density function \(v(x)\) and being independent of \(S_n\). Denote by \(f_n(x)\) the density of \((S_n+ \nu)/ \sqrt n\) and set \(\varphi_n(x) =\int \varphi(x-y/ \sqrt n) v(y)dy\), where \(\varphi\) is the density of the standard normal distribution. It was proved by V. V. Yurinskij that there exists a density \(v(x)\) and a constant \(C\) such that \(\int|f_n(x) -\varphi_n(x) |dx \leq C\gamma/ \sqrt n\), provided that \(\gamma= E|\xi |^3 <\infty\). In the proof of Yurinskij, the characteristic function \(\widehat v(t)\) of the chosen density \(v(x)\) vanishes when \(|t|\geq c_0/ \gamma\) for some constant \(c_0\). In this article, it is proved that the characteristic function \(\widehat v(t)\) of the density \(v(x)\) must vanish when \(|t|\geq c_0/ \gamma\) for \(c_0= 2\pi\), if the above inequality is true for all possible random variables \(\xi\), with \(\gamma\geq 7^{3/2}3^{-5/2}\).