An estimate of the speed of convergence in the multidmensional central limit theorem without moment hypotheses

Let $X_1,\dots,X_n$ ($n\ge1$) be independent random vectors in $R_d$, $b$b be a vector in $R_d$. For an arbitrary Borel set $A\subset R_d$ we set
\begin{gather*} P_n(A)=P\{X_1+\dots+X_n-b\in A\}, \\ \Delta_n(A)=|P_n(a)-\Phi(A)|, \end{gather*}
where $\Phi(A)$ is the probability function of a standard normal vector in $R_d$. In this note are obtained estimates for $\Delta_n(A)$, where $A$ belongs to the class of convex Borel sets in $R_d$.