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Let $X_1,\dots, X_n,\dots$ be i.i.d.\ $d$-dimensional random vectors withcommon 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 istaken 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 isconcentrated on a hyperplane that does not contain the origin. Clearly, forsuch $F$ $$ \rho_{\mathcal{C}_d}(F^n, F^{n+1}) = 1. $$ A similar result for theProkhorov 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 $.

Estimates of stability with respect to the number of summands for distributions of successive sums of independent identically distributed vectors