Berry Esseen inequality for multidimensional distributions - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T05:38:51Z http://mathoverflow.net/feeds/question/110960 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/110960/berry-esseen-inequality-for-multidimensional-distributions Berry Esseen inequality for multidimensional distributions blober 2012-10-29T05:02:54Z 2012-10-29T17:28:39Z <p>The classical Berry-Esseen theorem asserts that if $f$ and $g$ are the characteristic functions of two distribution functions $F(t)$ and $G(t)$ respectively then with $T$ arbitrary $$\sup_{t \in \mathbb{R}} |F(t) - G(t)| \ll \frac{1}{T} + \int_{-T}^{T} \bigg | \frac{f(t) - g(t)}{t} \bigg | dt$$ provided that one of the $F$ or $G$ is in Schwartz class (say). Is there a generalization of this inequality for distribution functions in $\mathbb{R}^k$, with $k = 2$ specifically? Precisely, I'm looking for a bound for $$\sup_{\mathcal{R}} |\mathbb{P}(X \in \mathcal{R}) - \mathbb{P}(Y \in \mathcal{R}) |$$ in terms of the characteristic functions of $X$ and $Y$, with $X,Y$ random variables in $\mathbb{R}^2$, and $\mathcal{R}$ rectangles in $\mathbb{R}^2$. </p> http://mathoverflow.net/questions/110960/berry-esseen-inequality-for-multidimensional-distributions/110984#110984 Answer by Ryan O'Donnell for Berry Esseen inequality for multidimensional distributions Ryan O'Donnell 2012-10-29T11:13:25Z 2012-10-29T11:13:25Z <p>The sharpest multidimensional Berry--Esseen Theorem I know is due to Bentkus and appears in the paper "A Lyapunov type bound in ${\mathbb R}^d$".</p> <p><a href="http://epubs.siam.org/doi/abs/10.1137/S0040585X97981123" rel="nofollow">http://epubs.siam.org/doi/abs/10.1137/S0040585X97981123</a></p> <p>It does not use the characteristic function, though.</p> http://mathoverflow.net/questions/110960/berry-esseen-inequality-for-multidimensional-distributions/111014#111014 Answer by Mark Meckes for Berry Esseen inequality for multidimensional distributions Mark Meckes 2012-10-29T17:28:39Z 2012-10-29T17:28:39Z <p>There are many results along those lines in Bhattacharya and Rao, <em>Normal Approximation and Asymptotic Expansions</em>.</p>