An extension of the S. N. Bernstein inequalities to multidimensional distributions

Yu. V. Prokhorov

Teoriâ veroâtnostej i ee primeneniâ, Tome 13 (1968) no. 2, pp. 266-274

Résumé

Let $X_1,\dots,X_n,\dots$ be a sequence of identically distributed independent random vectors in $R^m$ and $$ Y_n=\frac{X_1+\dots+X_n}{\sqrt n}, $$ Ir$\mathbf EX_j=0$, $|X_j|\le L$ and $n\ge m$, then $$ \mathbf P\{|Y_n|\ge r\}\le Ce^{-\frac{kr^2}{L^2}} $$ where $$ c\le1+\frac{e^{5/12}}{\pi/\sqrt2},\quad k\ge\frac1{8e^2}. $$