On a Multidimensional Version of the Kolmogorov Uniform Limit Theorem

E. L. Presman

Teoriâ veroâtnostej i ee primeneniâ, Tome 18 (1973) no. 2, pp. 396-402

Résumé

It is proved, that, for any $k$, there exist such a constant $c(k)$, that for any distribution function $F=F(x)$ in $R^k$, one can find a sequence of the vectors $\{a_n\}$ for which $$ \rho (F^n, E_{-na_{n}}\exp n (E_{a_n}F - E))<c(k)n^{-1/3} $$ where $\rho (F,g)=\sup_x |F(x)-G(x)|$, $F^n$ is the $n$-time convolution of $F$ with itself and $E_a$ is the distribution function corresponding to the unit mass at $a$.