An extension of the Erdős-Rényi new law of large numbers

Author:
Stephen A. Book

Journal:
Proc. Amer. Math. Soc. **48** (1975), 438-446

MSC:
Primary 60F15

DOI:
https://doi.org/10.1090/S0002-9939-1975-0380950-0

MathSciNet review:
0380950

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Abstract: If is the th partial sum of a sequence of independent, identically distributed random variables such that and for some nonempty interval of 's, then, for a wide range of positive numbers , Erdös and Rényi (1970) showed that converges with probability one to as , where is the maximum of the averages of the form for , and is a known constant depending on and the distribution of . The objective of the present article is to state and prove the Erdös-Rényi theorem for the ``averages'' of the form , where . This form of the Erdös-Rényi theorem arises from the extended form of the strong law of large numbers which asserts that, if for some , and , then converges with probability one to 0 as .

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1975-0380950-0

Keywords:
Strong limit theorems,
laws of large numbers,
large deviations,
moment-generating functions

Article copyright:
© Copyright 1975
American Mathematical Society