On a uniform bound for the rate of convergence in the multidi mensional local limit theorem for densities

Let $\{Xi\}$, $i\ge1$, be independent random vectors in $R^k$ with bounded densities $p_i(x)\le A_i<\infty$, such that $\mathbf EX_i=0$, $\mathbf E|X_i|^3=\beta_i<\infty$. If we denote $\sigma_i^2=\mathbf E|X_i|^2$, $B_n^2=\sum_{i=1}^n\sigma_i^2$, $K_n$ а matrix such that $Y_n=K_n\sum_{i=1}^nX_i$ has a unit covariance matrix, $u_n(x)$ and $\varphi(x)$ the densities of $Y_n$ and $k$-dimensional standard normal distribution respectively, then, under the assumptions (4) and (5), the relation (6) is true.