On Proximity of Distributions of Successive Sums with Respect to the Prokhorov Distance

Let $X, X_1,\dots, X_n,\dots$ be independent identically distributed $d$-dimensional random vectors with common distribution $F$. Let $F_{(n)}$ be the distribution of the normalized random vector $X/\sqrt{n}$. Then $(X_1+\dots+X_n)/\sqrt{n}$ has distribution $F_{(n)}^n$ (the power is understood in the convolution sense). Let $\pi(\,{\cdot}\,,{\cdot}\,)$ be the Prokhorov distance. We show that, for any $d$-dimensional distribution $F$, there exist $c_1(F)>0$ and $c_2(F)>0$ depending only on $F$ such that $\pi(F_{(n)}^n, F_{(n)}^{n+1})\leqslant c_1(F)/\sqrt n$ and $(F^n)\{A\} \le (F^{n+1})\{A^{c_2(F)}\}+c_2(F)/\sqrt{n}$, $(F^{n+1})\{A\} \leq (F^n)\{A^{c_2(F)}\}+c_2(F)/\sqrt{n}$ for each Borel set $A$ and for all natural numbers $n$ (here, $A^{\varepsilon}$ denotes the $\varepsilon$-neighborhood of a set $A$).