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

Let X, X₁, X₂,... 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 self-normalized partial sums S_n/V_n is established, where .

Mathematical Scientific Classification: 60F15.

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 non-degenerate distribution function F. For each n ≥ 1, the symbol S_n/V_n 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 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 well-known classical central limit theorem, Gine et al. [1] obtained the following self-normalized 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 σ² > 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 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 [[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 self-normalized 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 < ε₀ < 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.

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.

(ii) .

(iii) .

(iv) for α > 2.

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 ε₁ : = 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 < ε₂/(1 + ε₂), n_k = inf{n; D_n ≥ exp(k^1-η)}, then . Therefore

that is,

Since (1 - η)(1 + ε₂) > 1 from the definition of η, thus for any ε > 0, we have

By the Borel-Cantelli 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 non-negative, 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 non-negative, 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 non-negative, bounded Lipschitz function.

For any k ≥ 1, let

For any 1 ≤ k < j, note that and are independent and g(x) is a non-negative, 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 non-negative, 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.

The author declares that they have no competing interests.

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 (1103108-08).

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