- Open Access
Note on the strong convergence of a weighted sum
© Li et al.; licensee Springer. 2014
Received: 11 January 2014
Accepted: 21 April 2014
Published: 12 May 2014
In this paper, we consider an interesting weighted sums , , where is a sequence of independent and identically distributed random variables with , and the equivalence of the almost sure and complete convergence of is proved.
They also show that the sequence of arithmetic means of i.i.d. random variables converges completely to the expected value if the variance of the summands is finite. The converse was proved by Erdös [2, 3]. Furthermore, if completely, then the Borel-Cantelli lemma trivially implies that , almost sure convergence as . The converse statement is generally, or even ‘typically’, not true.
There are numerous publications in the literature studying the almost sure convergence and complete convergence for the weighted sums of a sequence of random variables. Gut  provided necessary and sufficient conditions for the complete convergence of the Cesáro means of i.i.d. random variables. Li et al.  obtained some results on complete convergence for weighted sums of independent random variables. Cuzick  proved a strong law for weighted sums of i.i.d. random variables. Miao and Xu  established a general result for the weighted sums of stationary sequence.
Throughout this paper, assume that is a sequence of independent and identically distributed random variables with . Chow and Lai  established the following result.
Theorem 1.1 [, Theorem 3]
a.e. for some (or equivalently for every) nonvoid sequence of real numbers (, ) such that , where .
The aim of this paper is to prove the following.
Theorem 1.2 We have if and only if completely.
and proved the equivalence of the almost sure and complete convergence of the sequence . On the one hand, because of the limitation of α in Theorem 1.1, we here only discuss the case of . On the other hand, in order to prove the equivalence of the almost sure and complete convergence of for the case , we need the exponential integrability of X, but from Theorem 1.1, this does not hold.
Let ; then we have the following.
2 Proofs of main results
Let denote the usual integer part of ‘ ⋅ ’ and assume that . The constant C in the proofs below depends only on the distribution of the underlying random variable X and may denote different quantities at different appearances.
2.1 Proof of Theorem 1.2
The proof of Theorem 1.2 can be derived from Lemma 2.2 and Lemma 2.3.
which yields the desired result. □
Similarly, we can obtain , then . The first statement now follows if we combine the two inequalities. The same technique yields the second statement. □
for , the random variables , are independent. Notice also that . So by the Borel-Cantelli lemma and the second statement of Lemma 2.2, we have , proving the lemma. □
2.2 Proof of Corollary 1.1
Replacing with , the corollary follows.
This work is supported by HASTIT (No. 2011HASTIT011), NSFC (No. 11001077), NCET (NCET-11-0945), and Plan For Scientific Innovation Talent of Henan Province (124100510014).
- Hsu PL, Robbins H: Complete convergence and the law of large number. Proc. Natl. Acad. Sci. USA 1947, 33: 25–31. 10.1073/pnas.33.2.25MathSciNetView ArticleGoogle Scholar
- Erdös P: On a theorem of Hsu and Robbins. Ann. Math. Stat. 1949, 20: 286–291. 10.1214/aoms/1177730037View ArticleGoogle Scholar
- Erdös P: Remark on my paper ‘On a theorem of Hsu and Robbins’. Ann. Math. Stat. 1950, 21: 138. 10.1214/aoms/1177729897View ArticleGoogle Scholar
- Gut A: Complete convergence and Cesáro summation for i.i.d. random variables. Probab. Theory Relat. Fields 1993, 97: 169–178. 10.1007/BF01199318MathSciNetView ArticleGoogle Scholar
- Li DL, Rao MB, Jiang TF, Wang XC: Complete convergence and almost sure convergence of weighted sums of random variables. J. Theor. Probab. 1995, 8: 49–76. 10.1007/BF02213454MathSciNetView ArticleGoogle Scholar
- Cuzick J: A strong law for weighted sums of i.i.d. random variables. J. Theor. Probab. 1995, 8: 625–641. 10.1007/BF02218047MathSciNetView ArticleGoogle Scholar
- Miao Y, Xu SF: Almost sure convergence of weighted sum. Miskolc Math. Notes 2013, 14: 173–181.MathSciNetGoogle Scholar
- Chow YS, Lai TL: Limiting behavior of weighted sums of independent random variables. Ann. Probab. 1973, 1: 810–824. 10.1214/aop/1176996847MathSciNetView ArticleGoogle Scholar
- Wu WB: On the strong convergence of a weighted sum. Stat. Probab. Lett. 1999, 1: 19–22.View ArticleGoogle Scholar
- Chow YS, Teicher H: Probability Theory. 3rd edition. Springer, New York; 1978.View ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.