Estimates of stability with respect to the number of summands for distributions of successive sums of i.i.d. vectors

Let $X_1, X_2,\dots$ be i.i.d. random vectors in $\mathbf R^d$ with distribution $F$. Then $S_n = X_1+\dots+X_n$ has distribution $F^n$ (degree is understood in the sense of convolutions). Let $ \rho(F,G) = \sup_A |F\{A\} - G\{A\}| $, where the supremum is taken over all convex subsets of $\mathbf R^d$. For any nontrivial distribution $F$ there is $c_1(F)$ such that $ \rho(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(F^n, F^{n+1}) = 1 $. A similar result for the Prokhorov distance is also formulated. For any $d$-dimensional distribution $F$ there is $c_2(F)$ such that $ (F^n)\{A\}\le (F^{n+1})\{A^{c_2(F)}\}+\frac{c_2(F)}{\sqrt{n}}$ and $(F^{n+1})\{A\}\leq (F^n)\{A^{c_2(F)}\}+\frac{c_2(F)} {\sqrt{n}} $ for all Borel set $ A $ and all natural $n$. Here $A^{\varepsilon }$ is $ \varepsilon $-neighborhood of the set $ A $.