An improved result in almost sure central limit theorem for self-normalized products of partial sums
© Wu and Chen; licensee Springer 2013
Received: 16 November 2012
Accepted: 5 March 2013
Published: 27 March 2013
Let be a sequence of independent and identically distributed random variables in the domain of attraction of the normal law. A universal result in an almost sure limit theorem for the self-normalized products of partial sums is established.
Keywordsdomain of attraction of the normal law self-normalized partial sums almost sure central limit theorem
Here and in the sequel, is a standard normal random variable, and denotes the convergence in distribution. We say that satisfies the central limit theorem (CLT).
In contrast to the well-known classical central limit theorem, Gine et al.  obtained the following self-normalized version of the central limit theorem: as if and only if (2) holds.
with and ; here and in the sequel, I denotes an indicator function, and is the standard normal distribution function. Some ASCLT results for partial sums were obtained by Lacey and Philipp , Ibragimov and Lifshits , Miao , Berkes and Csáki , Hörmann , Wu [18, 19]. Gonchigdanzan and Rempala  gave ASCLT for products of partial sums. Huang and Pang , Wu , and Zhang and Yang  obtained ASCLT results for self-normalized version.
Under mild moment conditions, ASCLT follows from the ordinary CLT, but in general, the validity of ASCLT is a delicate question of a totally different character as CLT. The difference between CLT and ASCLT lies in the weight in ASCLT.
The terminology of summation procedures (see, e.g., Chandrasekharan and Minakshisundaram , p.35) shows that the larger the weight sequence in (5) is, the stronger the relation becomes. By this argument, one should also expect to get stronger results if we use larger weights. It would be of considerable interest to determine the optimal weights.
On the other hand, by Theorem 1 of Schatte , (5) fails for weight . The optimal weight sequence remains unknown.
The purpose of this paper is to study and establish the ASCLT for self-normalized products of partial sums of random variables in the domain of attraction of the normal law. We show that the ASCLT holds under a fairly general growth condition on , .
In the following, we assume that is a sequence of i.i.d. positive random variables in the domain of attraction of the normal law with . Let , , , for . denotes . The symbol c stands for a generic positive constant which may differ from one place to another.
Our theorem is formulated in a general setting.
Here and in the sequel, F is the distribution function of the random variable .
By the terminology of summation procedures, we have the following corollary.
Corollary 1.2 Theorem 1.1 remains valid if we replace the weight sequence by such that , .
Remark 1.3 Our results give substantial improvements for weight sequence in Theorem 1.1 obtained by Zhang and Yang .
Remark 1.4 If X is in the domain of attraction of the normal law, then for . On the contrary, if , then X is in the domain of attraction of the normal law. Therefore, the class of random variables in Theorem 1.1 is of very broad range.
Remark 1.5 Essentially, the problem whether Theorem 1.1 holds for remains open.
Furthermore, the following three lemmas will be useful in the proof, and the first is due to Csörgo et al. .
X is in the domain of attraction of the normal law.
- (iv)for .
- (v)is a slowly varying function at ∞.
where and are defined by (6).
from the arbitrariness of ε.
from , i.e., (9) holds. This completes the proof of Lemma 2.2. □
where and are defined by (6) and f is a non-negative, bounded Lipschitz function.
from the Toeplitz lemma.
for any which is a non-negative, bounded Lipschitz function.
By Lemma 2.2, (19) holds.
By Lemma 2.2, (17) holds.
By Lemma 2.2, (18) holds. This completes the proof of Lemma 2.3. □
from , is a slowly varying function at ∞, and .
for any and .
This, combining with (16), (28) and the arbitrariness of β in (28), (24), holds.
Hence, (25) holds.
Hence, (26) holds. By similar methods used to prove (26), we can prove (27). This completes the proof of Theorem 1.1. □
Qunying Wu, Professor, Doctor, working in the field of probability and statistics.
The authors are very grateful to the referees and the editors for their valuable comments and some helpful suggestions that improved the clarity and readability of the paper. Supported by the National Natural Science Foundation of China (11061012), and the support Program of the Guangxi China Science Foundation (2012GXNSFAA053010).
- Arnold BC, Villaseñor JA: The asymptotic distribution of sums of records. Extremes 1998, 1(3):351–363.View ArticleGoogle Scholar
- Berkes I, Csáki E: A universal result in almost sure central limit theory. Stoch. Process. Appl. 2001, 94: 105–134. 10.1016/S0304-4149(01)00078-3View ArticleGoogle Scholar
- Billingsley P: Convergence of Probability Measures. Wiley, New York; 1968.Google Scholar
- Brosamler GA: An almost everywhere central limit theorem. Math. Proc. Camb. Philos. Soc. 1988, 104: 561–574. 10.1017/S0305004100065750MathSciNetView ArticleGoogle Scholar
- Chandrasekharan K, Minakshisundaram S: Typical Means. Oxford University Press, Oxford; 1952.Google Scholar
- Csörgo M, Szyszkowicz B, Wang QY: Donsker’s theorem for self-normalized partial sums processes. Ann. Probab. 2003, 31(3):1228–1240. 10.1214/aop/1055425777MathSciNetView ArticleGoogle Scholar
- Gine E, Götze F, Mason DM: When is the Student t -statistic asymptotically standard normal? Ann. Probab. 1997, 25: 1514–1531.MathSciNetView ArticleGoogle Scholar
- Gonchigdanzan K, Rempala G: A note on the almost sure central limit theorem for the product of partial sums. Appl. Math. Lett. 2006, 19: 191–196. 10.1016/j.aml.2005.06.002MathSciNetView ArticleGoogle Scholar
- Hörmann S: Critical behavior in almost sure central limit theory. J. Theor. Probab. 2007, 20: 613–636. 10.1007/s10959-007-0080-3View ArticleGoogle Scholar
- Huang SH, Pang TX: An almost sure central limit theorem for self-normalized partial sums. Comput. Math. Appl. 2010, 60: 2639–2644. 10.1016/j.camwa.2010.08.093MathSciNetView ArticleGoogle Scholar
- Ibragimov IA, Lifshits M: On the convergence of generalized moments in almost sure central limit theorem. Stat. Probab. Lett. 1998, 40: 343–351. 10.1016/S0167-7152(98)00134-5MathSciNetView ArticleGoogle Scholar
- Lacey MT, Philipp W: A note on the almost sure central limit theorem. Stat. Probab. Lett. 1990, 9: 201–205. 10.1016/0167-7152(90)90056-DMathSciNetView ArticleGoogle Scholar
- Miao Y: Central limit theorem and almost sure central limit theorem for the product of some partial sums. Proc. Indian Acad. Sci. Math. Sci. 2008, 118(2):289–294. 10.1007/s12044-008-0021-9MathSciNetView ArticleGoogle Scholar
- Pang TX, Lin ZY, Hwang KS: Asymptotics for self-normalized random products of sums of i.i.d. random variables. J. Math. Anal. Appl. 2007, 334: 1246–1259. 10.1016/j.jmaa.2006.12.085MathSciNetView ArticleGoogle Scholar
- Peligrad M, Shao QM: A note on the almost sure central limit theorem for weakly dependent random variables. Stat. Probab. Lett. 1995, 22: 131–136. 10.1016/0167-7152(94)00059-HMathSciNetView ArticleGoogle Scholar
- Rempala G, Wesolowski J: Asymptotics for products of sums and U -statistics. Electron. Commun. Probab. 2002, 7: 47–54.MathSciNetView ArticleGoogle Scholar
- Schatte P: On strong versions of the central limit theorem. Math. Nachr. 1988, 137: 249–256. 10.1002/mana.19881370117MathSciNetView ArticleGoogle Scholar
- Wu QY: Almost sure limit theorems for stable distribution. Stat. Probab. Lett. 2011, 81(6):662–672. 10.1016/j.spl.2011.02.003View ArticleGoogle Scholar
- Wu QY: An almost sure central limit theorem for the weight function sequences of NA random variables. Proc. Indian Acad. Sci. Math. Sci. 2011, 121(3):369–377. 10.1007/s12044-011-0036-5MathSciNetView ArticleGoogle Scholar
- Wu QY: Almost sure central limit theory for products of sums of partial sums. Appl. Math. J. Chin. Univ. Ser. B 2012, 27(2):169–180.View ArticleGoogle Scholar
- Wu QY: A note on the almost sure limit theorem for self-normalized partial sums of random variables in the domain of attraction of the normal law. J. Inequal. Appl. 2012., 2012: Article ID 17. doi:10.1186/1029–242X-2012–17Google Scholar
- Ye DX, Wu QY: Almost sure central limit theorem for product of partial sums of strongly mixing random variables. J. Inequal. Appl. 2011., 2011: Article ID 576301Google Scholar
- Zhang Y, Yang XY: An almost sure central limit theorem for self-normalized products of sums of i.i.d. random variables. J. Math. Anal. Appl. 2011, 376: 29–41. 10.1016/j.jmaa.2010.10.021MathSciNetView ArticleGoogle Scholar
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