Summary
We show that a Banach space valued random variableX such that\(\mathop {\lim }\limits_{x \to \infty } t^2 \mathbb{P}\left\{ {\left\| X \right\| > t} \right\} = 0\) satisfies the central limit theorem if and only if the following criterion on small balls is fulfilled:
Along the line of ideas used in the proof of this characterization, we present in addition a result obtained with J. Zinn on an almost sure randomized central limit theorem: if (and only if)\(\mathbb{E}\left\| X \right\|^2< \infty\), in order thatX satisfy the central limit theorem, it is necessary and sufficient that for almost every ω the sequence\(\left( {\sum\limits_{i = 1}^n {g_i X_i (\omega )} /\sqrt {n)} _{n \in \mathbb{N}} } \right)\) converges in distribution, where(g i ) i∈ℕ denotes an orthogaussian sequence independent of(X i ) i∈ℕ.
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Ledoux, M., Talagrand, M. Un critère sur les petites boules dans le théorème limite central. Probab. Th. Rel. Fields 77, 29–47 (1988). https://doi.org/10.1007/BF01848129
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DOI: https://doi.org/10.1007/BF01848129