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

Let $X_1,\dots ,X_n,\dots$ be i.i.d.\ $d$-dimensional random vectors with common distribution $F$. Then $S_n=X_1+\cdots +X_n$ has distribution $F^n$ (degree is understood in the sense of convolution). Let $${\rho }_{{\mathcal{C}}_d}(F,G)=\underset{A}{\sup }\mathrm{\mid }F\{A\}-G\{A\}\mathrm{\mid },$$ 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 $${\rho }_{{\mathcal{C}}_d}(F^n,F^{n+1})\le \frac{c_1(F)}{\sqrt{n}}$$ 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 $A^{\epsilon }$ is $\epsilon$-neighborhood of the set $A$.