Probability Theory and Related Fields

, Volume 102, Issue 1, pp 1-17

On the almost sure central limit theorem and domains of attraction

  • István BerkesAffiliated withMathematical Institute of the Hungarian Academy of Sciences

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We give necessary and sufficient criteria for a sequence (X n) of i.i.d. r.v.'s to satisfy the a.s. central limit theorem, i.e.,
$$\mathop {\lim }\limits_{N \to \infty } \frac{1}{{\log N}}\sum\limits_{k \leqslant N} {\frac{1}{k}I} \left\{ {\frac{{S_k }}{{a_k }} - b_k< x} \right\} = \phi (x)\,\,\,\,{\text{a}}{\text{.s}}{\text{.}}\,\,{\text{for}}\,{\text{all}}\,x$$
for some numerical sequences (a n), (b n) whereS n=X 1+...+X n andI denotes indicator function. Our method leads also to new results on the limit distributional behavior ofS n/an−bn along subsequences (“partial attraction”), as well as to necessary and sufficient criteria for averaged versions of the central limit theorem such as
$$\mathop {\lim }\limits_{N \to \infty } \frac{1}{N}\sum\limits_{k \leqslant N} P \left( {\frac{{S_k }}{{a_k }} - b_k< x} \right) = \phi (x)\,\,\,\,\,{\text{for}}\,{\text{all}}\,x.$$

Mathematics Subject Classification (1991)

60F05 60F15