Sur les déviations modérées des sommes de variables aléatoires vectorielles indépendantes de même loi. (On moderate deviations of sums of independent and identically distributed vector valued random variables)
Let \(X\) be a Borel random variable with values in a real separable Banach space \(B\) such that \(Ef(X)=0\) and \(Ef^ 2(X)<\infty\), \(\forall f\in B'\). Let \(\{X_ k,\;k\geq 1\}\) be a sequence of independent copies of \(X\) and set for each \(n\geq 1\), \(S_ n=X_ 1+\dots +X_ n\). Put \(I(x)=| x|^ 2/2\) or \(+\infty\) according as \(x\) belongs or not to reproducing kernel Hilbert space \((H,|\cdot|)\) associated to the covariance structure of the distribution of \(X\). Suppose that \(\{b_ n,\;n\geq 1\}\) is a sequence of positive numbers such that \(b_ n/\sqrt {n}\to\infty\) and \(b_ n/n\to 0\) as \(n\to \infty\). It is proved that for every closed set \(F\) of \(B\) \[ \limsup_{n\to \infty}nb^{-2}_ n\log P\{S_ n/b_ n\in F\}\leq -\inf_{x\in F}I(x) \] if and only if (i) \(S_ n/b_ n\to 0\), in probability as \(n\to\infty\), (ii) there exists \(M>0\) such that for any \(u>0\) \[ \limsup_{n\to\infty}nb^{-2}_ n\log(nP[\| X\| > ub_ n])\leq -u^ 2/M. \]
covariance structure of the distribution
scientific article; zbMATH DE number 57542