Abstract
Let X, X_{1}, X_{2},... be a sequence of independent and identically distributed random variables in the domain of attraction of a normal distribution. A universal result in almost sure limit theorem for the selfnormalized partial sums S_{n}/V_{n }is established, where .
Mathematical Scientific Classification: 60F15.
Keywords:
domain of attraction of the normal law; selfnormalized partial sums; almost sure central limit theorem1. Introduction
Throughout this article, we assume {X, X_{n}}_{n ∈ ℕ }is a sequence of independent and identically distributed (i.i.d.) random variables with a nondegenerate distribution function F. For each n ≥ 1, the symbol S_{n}/V_{n }denotes selfnormalized partial sums, where . We say that the random variable X belongs to the domain of attraction of the normal law, if there exist constants a_{n }> 0, b_{n }∈ ℝ such that
where is the standard normal random variable. We say that {X_{n}}_{n∈ℕ }satisfies the central limit theorem (CLT).
It is known that (1) holds if and only if
In contrast to the wellknown classical central limit theorem, Gine et al. [1] obtained the following selfnormalized version of the central limit theorem: as n → ∞ if and only if (2) holds.
Brosamler [2] and Schatte [3] obtained the following almost sure central limit theorem (ASCLT): Let {X_{n}}_{n∈ℕ }be i.i.d. random variables with mean 0, variance σ^{2 }> 0 and partial sums S_{n}. Then
with d_{k }= 1/k and , where I denotes an indicator function, and Φ(x) is the standard normal distribution function. Some ASCLT results for partial sums were obtained by Lacey and Philipp [4], Ibragimov and Lifshits [5], Miao [6], Berkes and Csáki [7], Hörmann [8], Wu [9,10], and Ye and Wu [11]. Huang and Zhang [12] and Zhang and Yang [13] obtained ASCLT results for selfnormalized 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 [[14], p. 35]) shows that the large the weight sequence {d_{k}; k ≥ 1} in (3) is, the stronger the relation becomes. By this argument, one should also expect to get stronger results if we use larger weights. And it would be of considerable interest to determine the optimal weights.
On the other hand, by the Theorem 1 of Schatte [3], Equation (3) fails for weight d_{k }= 1. The optimal weight sequence remains unknown.
The purpose of this article is to study and establish the ASCLT for selfnormalized partial sums of random variables in the domain of attraction of the normal law, we will show that the ASCLT holds under a fairly general growth condition on d_{k }= k^{1 }exp(ln k)^{α}), 0 ≤ α < 1/2.
Our theorem is formulated in a more general setting.
Theorem 1.1. Let {X, X_{n}}_{n∈ℕ }be a sequence of i.i.d. random variables in the domain of attraction of the normal law with mean zero. Suppose 0 ≤ α < 1/2 and set
then
By the terminology of summation procedures, we have the following corollary.
Corollary 1.2. Theorem 1.1 remain valid if we replace the weight sequence {d_{k}}_{k∈ℕ }by any such that .
Remark 1.3. Our results not only give substantial improvements for weight sequence in theorem 1.1 obtained by Huang [12]but also removed the condition , 0 < ε_{0 }< 1 in theorem 1.1 of [12].
Remark 1.4. If , then X is in the domain of attraction of the normal law. Therefore, the class of random variables in Theorems 1.1 is of very broad range.
Remark 1.5. Essentially, the open problem should be whether Theorem 1.1 holds for 1/2 ≤ α < 1 remains open.
2. Proofs
In the following, a_{n }~ b_{n }denotes lim_{n→∞ }a_{n}/b_{n }= 1. The symbol c stands for a generic positive constant which may differ from one place to another.
Furthermore, the following three lemmas will be useful in the proof, and the first is due to [15].
Lemma 2.1. Let X be a random variable with , and denote . The following statements are equivalent:
(i) X is in the domain of attraction of the normal law.
Lemma 2.2. Let {ξ, ξ_{n}}_{n∈ℕ }be a sequence of uniformly bounded random variables. If exist constants c > 0 and δ > 0 such that
then
where d_{k }and D_{n }are defined by (4).
Proof. Since
By the assumption of Lemma 2.2, there exists a constant c > 0 such that ξ_{k} ≤ c for any k. Noting that , we have exp(ln^{α }x), α < 1 is a slowly varying function at infinity. Hence,
By (6),
On the other hand, if α = 0, we have d_{k }= e/k, D_{n }~ e ln n, hence, for sufficiently large n,
If α > 0, note that
This implies
Thus combining ξ_{k} ≤ c for any k,
Since α < 1/2 implies (1  2α)/(2α) > 0 and ε_{1 }: = 1/(2α)  1 > 0. Thus, for sufficiently large n, we get
Let . Combining (8)(12), for sufficiently large n, we get
By (11), we have D_{n+1 }~ D_{n}. Let 0 < η < ε_{2}/(1 + ε_{2}), n_{k }= inf{n; D_{n }≥ exp(k^{1η})}, then . Therefore
that is,
Since (1  η)(1 + ε_{2}) > 1 from the definition of η, thus for any ε > 0, we have
By the BorelCantelli lemma,
Now for n_{k }< n ≤ n_{k+1}, by ξ_{k} ≤ c for any k,
from . I.e., (7) holds. This completes the proof of Lemma 2.2.
Let , b = inf{x ≥ 1; l(x) > 0} and
By the definition of η_{j}, we have and jl(η_{j } ε) > (η_{j } ε)^{2 }for any ε > 0. It implies that
For every 1 ≤ i ≤ n, let
Lemma 2.3. Suppose that the assumptions of Theorem 1.1 hold. Then
where d_{k }and D_{n }are defined by (4) and f is a nonnegative, bounded Lipschitz function.
Proof. By the cental limit theorem for i.i.d. random variables and as n → ∞ from , Lemma 2.1 (iii), and (13), it follows that
where denotes the standard normal random variable. This implies that for any g(x) which is a nonnegative, bounded Lipschitz function
Hence, we obtain
from the Toeplitz lemma.
On the other hand, note that (14) is equivalent to
from Theorem 7.1 of [16] and Section 2 of [17]. Hence, to prove (14), it suffices to prove
for any g(x) which is a nonnegative, bounded Lipschitz function.
For any k ≥ 1, let
For any 1 ≤ k < j, note that and are independent and g(x) is a nonnegative, bounded Lipschitz function. By the definition of η_{j}, we get,
By Lemma 2.2, (17) holds.
Now we prove (15). Let
It is known that I(A ∪ B)  I(B) ≤ I(A) for any sets A and B, then for 1 ≤ k < j, by Lemma 2.1 (ii) and (13), we get
Hence
By Lemma 2.2, (15) holds.
Finally, we prove (16). Let
For 1 ≤ k < j,
By Lemma 2.2, (16) holds. This completes the proof of Lemma 2.3.
Proof of Theorem 1.1. For any given 0 < ε < 1, note that
and
Hence, to prove (5), it suffices to prove
by the arbitrariness of ε > 0.
Firstly, we prove (19). Let 0 < β < 1/2 and h(·) be a real function, such that for any given x ∈ ℝ,
By , Lemma 2.1 (iii) and (13), we have
This, combining with (14), (23) and the arbitrariness of β in (23), (19) holds.
By (15), (18) and the Toeplitz lemma,
That is (20) holds.
Now we prove (21). For any μ > 0, let f be a nonnegative, bounded Lipschitz function such that
Form is i.i.d., Lemma 2.1 (iv), and (13),
Therefore, from (16) and the Toeplitz lemma,
Hence, (21) holds. By similar methods used to prove (21), we can prove (22). This completes the proof of Theorem 1.1.
Competing interests
The author declares that they have no competing interests.
Acknowledgements
The author was 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. This work was supported by the National Natural Science Foundation of China (11061012), the project supported by program to Sponsor Teams for Innovation in the Construction of Talent Highlands in Guangxi Institutions of Higher Learning ([2011]47), and the support program of Key Laboratory of Spatial Information and Geomatics (110310808).
References

Gine, E, Götze, F, Mason, DM: When is the Student tstatistic asymptotically standard normal?. Ann Probab. 25, 1514–531 (1997)

Brosamler, GA: An almost everywhere central limit theorem. Math Proc Camb Philos Soc. 104, 561–574 (1988). Publisher Full Text

Schatte, P: On strong versions of the central limit theorem. Mathematische Nachrichten. 137, 249–256 (1988). Publisher Full Text

Lacey, MT, Philipp, W: A note on the almost sure central limit theorem. Statist Probab Lett. 9, 201–205 (1990). Publisher Full Text

Ibragimov, IA, Lifshits, M: On the convergence of generalized moments in almost sure central limit theorem. Stat Probab Lett. 40, 343–351 (1998). Publisher Full Text

Miao, Y: Central limit theorem and almost sure central limit theorem for the product of some partial sums. Proc Indian Acad Sci C Math Sci. 118(2), 289–294 (2008). Publisher Full Text

Berkes, I, Csáki, E: A universal result in almost sure central limit theory. Stoch Proc Appl. 94, 105–134 (2001). Publisher Full Text

Hörmann, S: Critical behavior in almost sure central limit theory. J Theoret Probab. 20, 613–636 (2007). Publisher Full Text

Wu, QY: Almost sure limit theorems for stable distribution. Stat Probab Lett. 281(6), 662–672 (2011)

Wu, QY: An almost sure central limit theorem for the weight function sequences of NA random variables. Proc Math Sci. 121(3), 369–377 (2011). Publisher Full Text

Ye, DX, Wu, QY: Almost sure central limit theorem of product of partial sums for strongly mixing. J Inequal Appl vol. 2011, 9 (2011) Article ID 576301,
Article ID 576301,
BioMed Central Full Text 
Huang, SH, Pang, TX: An almost sure central limit theorem for selfnormalized partial sums. Comput Math Appl. 60, 2639–2644 (2010). Publisher Full Text

Zhang, Y, Yang, XY: An almost sure central limit theorem for selfnormalized products of sums of i.i.d. random variables. J Math Anal Appl. 376, 29–41 (2011). Publisher Full Text

Chandrasekharan K, Minakshisundaram S (eds.): Typical Means. Oxford University Press, Oxford (1952)

Csörgo, M, Szyszkowicz, B, Wang, QY: Donsker's theorem for selfnormalized partial processes. Ann Probab. 31(3), 1228–1240 (2003). Publisher Full Text

Billingsley P (ed.): Convergence of Probability Measures. Wiley, New York (1968)

Peligrad, M, Shao, QM: A note on the almost sure central limit theorem for weakly dependent random variables. Stat Probab Lett. 22, 131–136 (1995). Publisher Full Text