Un critère sur les petites boules dans le théorème limite central. (A criterion on small balls in the central limit theorem)
We show that a Banach space valued random variable X such that \(\lim _{t\to \infty}t^ 2{\mathbb{P}}\{\| X\| >t\}=0\) satisfies the central limit theorem if and only if the following criterion on small balls is fulfilled: \[ for\quad each\quad \epsilon >0,\quad \liminf _{n\to \infty}{\mathbb{P}}\{\| S_ n/\sqrt{n}\| <\epsilon \}>0. \] Along the line of ideas used in the proof of this characterization, we present in addition a result obtained by the first author and J. Zinn on an almost sure randomized central limit theorem: if (and only if) \({\mathbb{E}}\| X\| ^ 2<\infty\), in order that X satisfies the central limit theorem, it is necessary and sufficient that for almost every \(\omega\), the sequence \((\sum ^{n}_{i=1}g_ iX_ i(\omega)/\sqrt{n})_{n\in {\mathbb{N}}}\) converges in distribution, where \((g_ i)_{i\in {\mathbb{N}}}\) denotes an orthogaussian sequence independent of \((X_ i)_{i\in {\mathbb{N}}}\).
scientific article; zbMATH DE number 4001075
scientific article