Generalization of some V. M. Zolotarev's result (Q2730573)

Let \(\xi_1,\xi_2,\ldots\) be a sequence of independent random variables with \(M\xi_{i}=0\), \(D\xi_{i}=\sigma^2_{i}\), the distribution function \(F_{i}(x)\), and the characteristic function \(f_{i}(t)\). Denote by \(\Phi(x)\) the distribution function of the standard normal law; by \(\Phi_{n}(x)\) the distribution function of the random variable \((\xi_1+\ldots+\xi_{n})/B_{n}\), \(B_{n}^2=\sigma_1^2+\ldots+\sigma_{n}^2\), \(\overline\sigma_{i}= \min (1,\sigma_{i})\), \(\overline B_{n}^2=\overline\sigma_1^2+\ldots+ \overline\sigma_{n}^2\), \(\rho_{n}=\sup_{x}|\Phi_{n}(x)-\Phi (x)|\). The main result is as follows. Let \(\theta_{i}\) be such that for some \(s\in [0,3 ]\) for all \(t\) we have NEWLINE\[NEWLINE|f_{i}(t)- e^{-t^2\sigma_{i}^2/2}|\leq \theta_{i}\min (|t|^{s}/m(s),|t|^{3}/6), \quad m(s)\geq 1.NEWLINE\]NEWLINE Then there exists a constant \(A(s)\) such that for \(n\geq 2\): NEWLINE\[NEWLINE\rho_{n}\leq (A(s)/\overline B_{n}b_{n}^{s+1})\max(\overline\theta_{n},\overline\theta_{n}^{p}),NEWLINE\]NEWLINE where NEWLINE\[NEWLINE\overline\theta_{n}=\sum_{i=1}^{n}\theta_{i}/\overline B_{n}^2, \quad b_{n}=\min_{1\leq i\leq n}\overline\sigma_{i}, \quad p=\begin{cases} 1, & \theta_{n}\geq 1, \\ \min(1,n/(sn+1)), & \theta_{n}<1.\end{cases}NEWLINE\]NEWLINE{}.