Kybernetika

Let $(X_n, n\ge 1), (\tilde{X}_n, n\ge 1)$ be two sequences of i.i.d. random vectors with values in $\R^k$ and $S_n=X_1+\cdots +X_n$, $\tilde{S}_n=\tilde{X}_1+\cdots +\tilde{X}_n$, $n\ge 1$. Assuming that $EX_1=E\tilde{X}_1$, $E|X_1|^2<\infty$, $E|\tilde{X}_1|^{k+2}<\infty$ and the existence of a density of $\tilde{X}_1$ satisfying the certain conditions we prove the following inequalities: $$\bv(S_n,\tilde{S}_n)\le c\;\max\big\{\bv(X_1,\tilde{X}_1), \zeta_2(X_1,\tilde{X}_1)\big\}, \quad n=1,2,\dots,$$ where $\bv$ and $\zeta_2$ are the total variation and Zolotarev's metrics, respectively.