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An improved result in almost sure central limit theorem for self-normalized products of partial sums
Journal of Inequalities and Applications volume 2013, Article number: 129 (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.
Let be a sequence of independent and identically distributed (i.i.d.) positive random variables with a non-degenerate distribution function and . For each , the symbol denotes self-normalized partial sums, where , . We say that the random variable X belongs to the domain of attraction of the normal law if there exist constants , such that
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).
It is known that (1) holds if and only if
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.
The limit theorem of products was initiated by Arnold and Villaseñor . Their result was generalized by Wu , Ye and Wu , and Rempala and Wesolowski  who proved that if is a sequence of i.i.d. positive and finite second moment random variables with , and the coefficient of variation , then
Recently Pang et al.  obtained the following self-normalized products of sums for i.i.d. sequences: Let be a sequence of i.i.d. positive random variables with , and assume that X is in the domain of attraction of the normal law. Then
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.
Theorem 1.1 Let be a sequence of i.i.d. positive random variables in the domain of attraction of the normal law with mean . Suppose and set
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. .
Lemma 2.1 Let X be a random variable with , and denote . The following statements are equivalent.
X is in the domain of attraction of the normal law.
is a slowly varying function at ∞.
Lemma 2.2 Let be a sequence of uniformly bounded random variables. If there exist constants and such that
where and are defined by (6).
By the assumption of Lemma 2.2, there exists a constant such that for any k. Noting that , we have that , is a slowly varying function at infinity. Hence,
On the other hand, if , we have , , and hence, for sufficiently large n,
If , then by , , for arbitrary small , there exists such that for , . Therefore
from the arbitrariness of ε.
Thus combining for any k,
Since implies and , thus for sufficiently large n, we get
Let , . Combining (10)-(12) and (14), for sufficiently large n, we get
By (13), we have . Let , , then , . Therefore
Since from the definition of η, thus for any , we have
By the Borel-Cantelli lemma,
Now, for , by for any k,
from , i.e., (9) holds. This completes the proof of Lemma 2.2. □
Let , and
By the definition of , we have and for any . It implies that
For every , let
Lemma 2.3 Suppose that the assumptions of Theorem 1.1 hold. Then
where and are defined by (6) and f is a non-negative, bounded Lipschitz function.
Proof By the central limit theorem for i.i.d. random variables and as from , it follows that
where denotes the standard normal random variable. This implies that for any which is a non-negative, bounded Lipschitz function,
Hence, we obtain
from the Toeplitz lemma.
On the other hand, note that (16) is equivalent to
for any which is a non-negative, bounded Lipschitz function.
For any , let
For any , note that and are independent and is a non-negative, bounded Lipschitz function. By the definition of , we get
By Lemma 2.2, (19) holds.
Now we prove (17). Let
It is known that for any sets A and B. Then for , by Lemma 2.1(ii) and (15), we get
By Lemma 2.2, (17) holds.
Finally, we prove (18). Let
By Lemma 2.2, (18) holds. This completes the proof of Lemma 2.3. □
Proof of Theorem 1.1 Let ; then (7) is equivalent to
Let , then and from Remark 1.4. Using the Marcinkiewicz-Zygmund strong large number law, we have
Hence let for any given , by for ,
from , is a slowly varying function at ∞, and .
Therefore, for any and almost every event ω, there exists such that for ,
Note that under the condition , ,
Thus, by (22) and (23), for any given , , we have for ,
Hence, to prove (21), it suffices to prove
for any and .
Firstly, we prove (24). Let , and let be a real function such that for any given ,
By , Lemma 2.1(iii) and (15), we have
This, combining with (16), (28) and the arbitrariness of β in (28), (24), holds.
By (17), (20) and the Toeplitz lemma,
Hence, (25) holds.
Now we prove (26). For any , let f be a non-negative, bounded Lipschitz function such that
From , is i.i.d., Lemma 2.1(iv), and (15),
Therefore, from (18) and the Toeplitz lemma,
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.
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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).
The authors declare that they have no competing interests.
QW conceived of the study and drafted, complete the manuscript. PC participated in the discussion of the manuscript. QW and PC read and approved the final manuscript.
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Wu, Q., Chen, P. An improved result in almost sure central limit theorem for self-normalized products of partial sums. J Inequal Appl 2013, 129 (2013). https://doi.org/10.1186/1029-242X-2013-129
- domain of attraction of the normal law
- self-normalized partial sums
- almost sure central limit theorem