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

Abstract: 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 $.