Some multivariate inequalities with applications

Summary Let <InlineEquation ID=IE"1"><EquationSource Format="TEX"><!CDATA<InlineEquation ID=IE"2"><EquationSource Format="TEX"><!CDATA<InlineEquation ID=IE"3"><EquationSource Format="TEX"><!CDATA<InlineEquation ID=IE"4"><EquationSource Format="TEX"><!CDATA<InlineEquation ID=IE"5"><EquationSource Format="TEX"><!CDATA<InlineEquation ID=IE"6"><EquationSource Format="TEX"><!CDATA<InlineEquation ID=IE"7"><EquationSource Format="TEX"><!CDATA<InlineEquation ID=IE"8"><EquationSource Format="TEX"><!CDATA<InlineEquation ID=IE"9"><EquationSource Format="TEX"><!CDATA$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>{\cal {X}}_{n} =(X_1,\ldots,X_n)$ be a random vector. Suppose that the random variables $(X_i)_{1\leq i\leq n}$ are stationary and fulfill a suitable dependence criterion. Let $f$ be a real valued function defined on $\mathbbm{R}^n$ having some regular properties. Let ${\cal {Y}}_{n}$ be a random vector, independent of ${\cal {X}}_{n}$, having independent and identically distributed components. We control $\left|\mathbbm{E}(f({\cal {X}}_{n}))-\mathbbm{E} (f({\cal {Y}}_{n}))\right|$. Suitable choices of the function $f$ yield, under minimal conditions, to rates of convergence in the central limit theorem, to some moment inequalities or to bounds useful for Poisson approximation. The proofs are derived from multivariate extensions of Taylor's formula and of the Lindeberg decomposition. In the univariate case and in the mixing setting the method is due to Rio (1995).