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

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., $$\mathop {\lim }\limits_{N \to \infty } \frac{1}{{\log N}}\sum\limits_{k \leqslant N} {\frac{1}{k}I\left\{ {\frac{{S_k }}{{a_k }} - b_k< x} \right\} = \phi (x)} a.s{\mathbf{ }}for{\mathbf{ }}all{\mathbf{ }}x$$ a.s. for allx whereS n =X 1+X 2+...+X n , (a n ) and (b n ) are numerical sequences andI denotes indicator function. By a recent result of Lacey and Philipp,(7) and Fisher,(6) and its converse obtained here, the a.s. central limit theorem holds with $$a_n = \sqrt n $$ iffEX 1 2 <+∞ which shows that for $$a_n = \sqrt n $$ the a.s. central limit theorem and the corresponding ordinary (weak) central limit theorem are equivalent. Our main results show that for general (a n ) the situation is radically different and paradoxical: the weak central limit theorem implies the a.s. central limit theorem but the converse is false and in fact the validity of the a.s. central limit theorem permits very irregular distributional behavior ofS n /a n −b n . We also show that the validity of the a.s. central limit theorem is closely connected with the limiting behavior of the classical fractionx 2(1−F(x)+F(−x))/∫|t|≤x t 2 dF(t) whereF is the distribution function ofX 1.