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

A. Yu. Zaitsev

Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 34, Tome 525 (2023), pp. 86-95

Résumé

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 $.