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On the almost sure (a.s.) central limit theorem for random variables with infinite variance

Authors:

Institution:

1. Mathematical Institute of the Hungarian Academy of Sciences, Reáltanoda u. 13-15, H-1053, Budapest, Hungary
2. Department of Mathematics, Groningen University, P.O. Box 800, NL-9700 AV, Groningen, The Netherlands

Abstract:

We give criteria for a sequence (X _n) of i.i.d.r.v.'s to satisfy the a.s. central limit theorem, i.e.,

$\mathop {\lim }\limits_{N \to \infty } \frac{1}{{\log N}}\sum\limits_{k \leqslant N} {\frac{1}{k}I\left\{ {\frac{{S_k }}{{a_k }} - b_k< x} \right\} = \phi (x)} a.s{\mathbf{ }}for{\mathbf{ }}all{\mathbf{ }}x$

Keywords:	Central limit theorem random variables with infinite variance
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