Summary
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.,
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
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Research supported by Hungarian National Foundation for Scientific Research, Grant No. 1905
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Berkes, I. On the almost sure central limit theorem and domains of attraction. Probab. Theory Relat. Fields 102, 1–17 (1995). https://doi.org/10.1007/BF01295218
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DOI: https://doi.org/10.1007/BF01295218