The Erdős-Rényi new law of large numbers for weighted sums

Author:
Stephen A. Book

Journal:
Proc. Amer. Math. Soc. **38** (1973), 165-171

MSC:
Primary 60F15

DOI:
https://doi.org/10.1090/S0002-9939-1973-0310946-4

MathSciNet review:
0310946

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: From the partial sums of the first random variables of a sequence of independent, identically distributed random variables, averages of the form can be constructed, one such average for each between 0 and , inclusive. If we denote by the maximum of those averages, then for a wide range of numbers , Erdös and Rényi (1970) proved that, as a.e. for , where is a constant depending only on , not on . The objective of the present article is to extend the Erdös-Rényi theorem to the case of weighted sums. The main theorem bears a relation to the law of large numbers for weighted sums of Jamison, Orey, and Pruitt (1965) similar to the one borne by the Erdös-Rényi theorem to the ordinary strong law of large numbers.

**[1]**R. R. Bahadur and R. Ranga Rao,*On deviations of the sample mean*, Ann. Math. Statist.**31**(1960), 1015-1027. MR**22**#8459. MR**0117775 (22:8549)****[2]**S. A. Book,*Large deviation probabilities for weighted sums*, Ann. Math. Statist.**43**(1972), 1221-1234. MR**0331486 (48:9819)****[3]**P. Erdös and A. Rényi,*On a new law of large numbers*, J. Analyse Math.**23**(1970), 103-111. MR**42**#6907. MR**0272026 (42:6907)****[4]**B. Jamison, S. Orey and W. Pruitt,*Convergence of weighted averages of independent random variables*, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete**4**(1965), 40-44. MR**31**#6268. MR**0182044 (31:6268)**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
60F15

Retrieve articles in all journals with MSC: 60F15

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1973-0310946-4

Keywords:
Strong limit theorems,
laws of large numbers,
large deviations,
weighted averages

Article copyright:
© Copyright 1973
American Mathematical Society