A multidimensional generalization of Esseen's inequality for distribution functions

N. G. Gamkrelidze

Teoriâ veroâtnostej i ee primeneniâ, Tome 22 (1977) no. 4, pp. 897-900

Résumé

Let $\xi$ and $\eta$ be $s$-dimensional random vectors with distribution functions $F(x)$, $G(x)$ and characteristic functions $f(t)$, $g(t)$ respectively. Theorem. {\it For arbitrary $T>0$, $$ \sup_x|F(x)-G(x)|\le 2\biggl[\frac{1}{(2\pi)^s}\int_{-T}^T|\Delta(t)|\,dt+ \sum_{k=1}^{s-1}\frac{1}{(2\pi)^{s-k}}\sum_{i(k)}\int_{-T}^T|\Delta_{i(k)}(t)|\,dt\biggr]+\frac{A}{T}C(s), $$ where $$ C(s)=\frac{24\ln 2}{\pi}+\frac{8s^{1/3}}{(2\pi\ln4/3)^{1/3}},\qquad A=\sup_x\frac{\partial G}{\partial x_1}+\dots+\sup_x\frac{\partial G}{\partial x_s} $$ and $\Delta(t)$, $\Delta_{i(k)}(t)$ are defined by} (3), $i(k)=\{i_1,\dots,i_k\}$ is an ordered sample from the sequence $(1,\dots,s)$.