Open Access
October, 1973 Limiting Behavior of Weighted Sums of Independent Random Variables
Y. S. Chow, T. L. Lai
Ann. Probab. 1(5): 810-824 (October, 1973). DOI: 10.1214/aop/1176996847


In this paper, we study weighted sums $\sum^n_{i=1} c_{n-i} X_i$ of i.i.d. zero-mean random variables $X_1, X_2, \cdots$, under the condition that the sequence $(c_n)$ is square summable. It is proved that such weighted sums are, with probability 1, of smaller order than $n^{1/\alpha}$ (respectively $\log n$, etc.) $\operatorname{iff} E|X_1|^\alpha < \infty$ (respectively $Ee^{t|X_1|} < \infty$ for all $t < \infty$, etc.). Certain analogs of the law of the iterated logarithm for such weighted sums are also obtained.


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Y. S. Chow. T. L. Lai. "Limiting Behavior of Weighted Sums of Independent Random Variables." Ann. Probab. 1 (5) 810 - 824, October, 1973.


Published: October, 1973
First available in Project Euclid: 19 April 2007

zbMATH: 0303.60025
MathSciNet: MR353426
Digital Object Identifier: 10.1214/aop/1176996847

Primary: 60J30
Secondary: 60F15

Keywords: coin tossing , double arrays , Exponential bounds , Khintchine inequality , Law of the iterated logarithm , Marcinkiewicz-Zygmund inequalities , Strong law of large numbers , success runs , symmetrization , weighted sums

Rights: Copyright © 1973 Institute of Mathematical Statistics

Vol.1 • No. 5 • October, 1973
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