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
where
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:
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
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
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
Remark 1.4. If
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
(i) X is in the domain of attraction of the normal law.
(ii)
(iii)
(iv)
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
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
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
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
Let
By the definition of η_{j}, we have
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
where
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
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
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
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).
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