On a necessary conditon for the Erdős-Rényi law of large numbers

Author:
Josef Steinebach

Journal:
Proc. Amer. Math. Soc. **68** (1978), 97-100

MSC:
Primary 60F15

MathSciNet review:
0461637

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Abstract: For a sequence of independent, identically distributed random variables with existing moment-generating function in some nondegenerate interval, Erdös and Rényi (1970) studied the maximum of the sample means , where . They showed that for a certain range of numbers a there exist positive constants such that with probability one. In the present paper it is shown that the existence of the moment-generating function is also a necessary condition, i.e. that for every positive constant *C*, if the moment-generating function does not exist for any positive number *t*.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1978-0461637-5

Keywords:
Erdös-Rényi law of large numbers,
large deviations,
moment-generating functions

Article copyright:
© Copyright 1978
American Mathematical Society