  Let $X_1,\dots, X_n,\dots$ be i.i.d.\ $d$-dimensional random vectors with
common distribution $F$. Then $S_n = X_1+\dots+X_n$ has distribution $F^n$
(degree is understood in the sense of convolution). Let $$
\rho_{\mathcal{C}_d}(F,G) = \sup_A |F\{A\} - G\{A\}|, $$ where the supremum is
taken over all convex subsets of $\mathbb R^d$. Basic result is as follows. For
any nontrivial distribution $F$ there is $c_1(F)$ such that $$
\rho_{\mathcal{C}_d}(F^n, F^{n+1})\leq \frac{c_1(F)}{\sqrt n} $$ for any
natural $n$. The distribution $F$ is called trivial if it is concentrated on a
hyperplane that does not contain the origin. Clearly, for such $F$ $$
\rho_{\mathcal{C}_d}(F^n, F^{n+1}) = 1. $$ A similar result for the Prokhorov
distance is also obtained. For any $d$-dimensional distribution~$F$ there is a
$c_2(F)>0$ that depends only on $F$ and such that \begin{multline}\nonumber
(F^n)\{A\}\le (F^{n+1})\{A^{c_2(F)}\}+\frac{c_2(F)}{\sqrt{n}} \text{and}\quad
(F^{n+1})\{A\}\leq (F^n)\{A^{c_2(F)}\}+\frac{c_2(F)} {\sqrt{n}} \end{multline}
for any Borel set $ A $ for all positive integers $n$. Here $A^{\varepsilon }$
is $ \varepsilon $-neighborhood of the set $ A $.
