Almost Sure Central Limit Theory for Self-Normalized Products of Sums of Partial Sums

Abstract

Let 𝑋,𝑋₁,𝑋₂,… be a sequence of independent and identically distributed random variables in the domain of attraction of a normal law. An almost sure limit theorem for the self-normalized products of sums of partial sums is established.

1. Introduction

Let {𝑋,𝑋_𝑛}_𝑛∈ℕ be a sequence of independent and identically distributed (i.i.d.) positive random variables with a nondegenerate distribution function and 𝔼𝑋=𝜇>0. For each 𝑛≥1, the symbol 𝑆_𝑛/𝑉_𝑛 denotes self-normalized partial sums, where 𝑆_𝑛=∑^𝑛_𝑖=1𝑋_𝑖 and 𝑉²_𝑛=∑^𝑛_𝑖=1(𝑋_𝑖−𝜇)². 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 that𝑆_𝑛−𝑏_𝑛𝑎_𝑛^𝑑⟶𝒩,as𝑛⟶∞, (1.1) here and in the sequel 𝒩 is a standard normal random variable. We say that {𝑋,𝑋_𝑛}_𝑛∈ℕ satisfies the central limit theorem (CLT).

It is known that (1.1) holds if and only iflim_𝑥→∞𝑥²ℙ(|𝑋|>𝑥)𝔼𝑋²𝐼(|𝑋|≤𝑥)=0. (1.2) 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: (𝑆_𝑛−𝔼𝑆_𝑛)/𝑉−→_𝑛^𝑑𝒩 as 𝑛→∞ if and only if (1.2) 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. Thenlim_𝑛→∞1𝐷_𝑛^𝑛_𝑘=1𝑑_𝑘𝐼𝑆_𝑘𝜎√𝑘<𝑥=Φ(𝑥)a.s.∀𝑥∈ℝ, (1.3) with 𝑑_𝑘=1/𝑘 and 𝐷_𝑛=∑^𝑛_𝑘=1𝑑_𝑘; here and in the sequel 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 Pang [12] and Zhang and Yang [13] obtained ASCLT results for self-normalized version. However, ASCLT results for self-normalized products of sums of partial sums have not been reported yet.

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], page 35) shows that the larger the weight sequence {𝑑_𝑘;𝑘≥1} in (1.3) 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], (1.3) fails for weight 𝑑_𝑘=1. The optimal weight sequence remains unknown.

The purpose of this paper is to study and establish the ASCLT for self-normalized products of sums of 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⁻¹exp((ln𝑘)^𝛼), 0≤𝛼<1/2.

In the following, we assume that {𝑋,𝑋_𝑛}_𝑛∈ℕ is a sequence of i.i.d. positive random variables in the domain of attraction of the normal law with 𝔼𝑋=𝜇>0 and define 𝑆_𝑘=∑^𝑘_𝑖=1𝑋_𝑖,𝑉_𝑘=∑^𝑘_𝑖=1(𝑋_𝑖−𝜇)²,𝑇_𝑘=∑^𝑘_𝑖=1𝑆_𝑖. Let 𝑏_𝑛,𝑘=∑^𝑛_𝑗=𝑘1/𝑗,𝑐_𝑛,𝑘=2∑^𝑛_𝑗=𝑘(𝑗+1−𝑘)/(𝑗(𝑗+1)), and 𝑑_𝑛,𝑘=(𝑛+1−𝑘)/(𝑛+1) for 1≤𝑘≤𝑛. 𝐼(𝐴) denotes the indicator function of set 𝐴, and 𝑎_𝑛∼𝑏_𝑛 denotes 𝑎_𝑛/𝑏_𝑛→1,𝑛→∞. The symbol 𝑐 stands for a generic positive constant, which may differ from one place to another. Let𝑙(𝑥)=𝔼𝑋−𝜇²𝐼{(𝑥)},𝜂𝑋−𝜇≤𝑥,𝑏=inf𝑥≥1;𝑙>0_𝑗𝑙(𝑠)=inf𝑠;𝑠≥𝑏+1,𝑠²≤1𝑗for𝑗≥1. (1.4)

By the definition of 𝜂_𝑗, we have 𝑗𝑙(𝜂_𝑗)≤𝜂²_𝑗 and 𝑗𝑙(𝜂_𝑗−𝜀)>(𝜂_𝑗−𝜀)² for any 𝜀>0. It implies that𝑛𝑙𝜂_𝑛∼𝜂²_𝑛,as𝑛→∞,𝜂_𝑛<𝑛+1. (1.5)

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. positive random variables in the domain of attraction of the normal law with 𝔼𝑋=𝜇>0. Suppose 𝑙𝜂_𝑛𝜂∼𝑙^𝑛ln𝑛. (1.6)
For 0≤𝛼<1/2, set 𝑑_𝑘=expln^𝛼𝑘𝑘,𝐷_𝑛=^𝑛_𝑘=1𝑑_𝑘. (1.7) Then lim_𝑛→∞1𝐷_𝑛^𝑛_𝑘=1𝑑_𝑘𝐼^𝑘_𝑗=1𝑇_𝑗𝑗𝑗+1𝜇/2^{𝜇/𝑉𝑘}≤𝑥=𝐹(𝑥)𝑎.𝑠., (1.8) for any 𝑥∈ℝ, where 𝐹 is the distribution function of the random variable exp(10/3𝒩).

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 {𝑑_𝑘}_𝑘∈ℕ by {𝑑^∗_𝑘}_𝑘∈ℕ such that 0≤𝑑^∗_𝑘≤𝑑_𝑘, ∑^∞_𝑘=1𝑑^∗_𝑘=∞.

Remark 1.3. If 𝔼𝑋²=𝜎²<∞, then 𝑋 is in the domain of attraction of the normal law and 𝑙(𝑥)→𝜎²,𝜂²_𝑛∼𝜎²𝑛, thus (1.6) holds. Therefore, the class of random variables in Theorems 1.1 is of very broad range.

Remark 1.4. Whether Theorem 1.1 holds for 1/2≤𝛼<1 remains open.

2. Proofs

Furthermore, the following four lemmas will be useful in the proof, and the first is due to Csörgo et al. [15].

Lemma 2.1. Let 𝑋 be a random variable with 𝔼𝑋=𝜇, and denote 𝑙(𝑥)=𝔼(𝑋−𝜇)²𝐼{|𝑋−𝜇|≤𝑥}. The following statements are equivalent: (i)𝑙(𝑥) is a slowly varying function at ∞;(ii)𝑋 is in the domain of attraction of the normal law;(iii)𝑥²ℙ(|𝑋−𝜇|>𝑥)=𝑜(𝑙(𝑥));(iv)𝑥𝔼(|𝑋−𝜇|𝐼(|𝑋−𝜇|>𝑥))=𝑜(𝑙(𝑥));(v)𝔼(|𝑋−𝜇|^𝛼𝐼(|𝑋−𝜇|≤𝑥))=𝑜(𝑥^𝛼−2𝑙(𝑥)) for 𝛼>2.

Lemma 2.2. Let {𝜉,𝜉_𝑛}_𝑛∈ℕ be a sequence of uniformly bounded random variables. If there exist constants 𝑐>0 and 𝛿>0 such that 𝔼𝜉_𝑘𝜉_𝑗𝑘≤𝑐𝑗^𝛿,for1≤𝑘<𝑗, (2.1) then lim_𝑛→∞1𝐷_𝑛^𝑛_𝑘=1𝑑_𝑘𝜉_𝑘=0𝑎.𝑠., (2.2) where 𝑑_𝑘 and 𝐷_𝑛 are defined by (1.7).

Proof. We can easily apply the similar arguments of (2.1) in Wu [16] to get Lemma 2.2, and we omit the details here.

The following Lemma 2.3 can be directly verified.

Lemma 2.3. (i) 𝑐_𝑛,𝑖=2(𝑏_𝑛,𝑖−𝑑_𝑛,𝑖);
(ii)^𝑛∑_𝑖=1𝑏²_𝑛,𝑖=2𝑛−𝑏_𝑛,1∼2𝑛; (iii)^𝑛∑_𝑖=1𝑑²_𝑛,𝑖=𝑛3−𝑛∼𝑛6(𝑛+1)3; (iv)^𝑛∑_𝑖=1𝑐²_𝑛,𝑖=10𝑛3−4𝑏_𝑛,1+10𝑛∼3(𝑛+1)10𝑛3.

For every 1≤𝑖≤𝑘≤𝑛, let𝑋_𝑘𝑖=𝑋_𝑖−𝜇𝐼𝑋_𝑖−𝜇≤𝜂_𝑘,𝑆_𝑘,𝑖=^𝑖_𝑗=1𝑐_𝑘,𝑗𝑋_𝑘𝑗,𝑉²_𝑘=^𝑘_𝑗=1𝑋²_𝑘𝑗. (2.3)

Lemma 2.4. Suppose that the assumptions of Theorem 1.1 hold. Then lim_𝑛→∞1𝐷_𝑛^𝑛_𝑘=1𝑑_𝑘𝐼𝑆_𝑘,𝑘−𝔼𝑆_𝑘,𝑘10𝑘𝑙𝜂_𝑘(𝑥)/3≤𝑥=Φa.s.forany𝑥∈ℝ, (2.4)lim_𝑛→∞1𝐷_𝑛^𝑛_𝑘=1𝑑_𝑘𝐼^𝑘_𝑖=1𝑋_𝑖−𝜇>𝜂_𝑘−𝔼𝐼^𝑘_𝑖=1𝑋_𝑖−𝜇>𝜂_𝑘=0a.s., (2.5)lim_𝑛→∞1𝐷_𝑛^𝑛_𝑘=1𝑑_𝑘𝑓𝑉²_𝑘𝑘𝑙𝜂_𝑘−𝔼𝑓𝑉²_𝑘𝑘𝑙𝜂_𝑘=0a.s., (2.6) where 𝑑_𝑘 and 𝐷_𝑛 are defined by (1.7) and 𝑓 is a nonnegative, bounded Lipschitz function.

Proof. By 𝔼(𝑋−𝜇)=0, Lemma 2.1(iv), we have 𝔼𝑋_𝑛𝑖≤𝔼𝑋−𝜇𝐼𝑋−𝜇>𝜂_𝑛=𝑜𝑙𝜂_𝑛𝜂_𝑛. (2.7) Thus, by (1.5) and Lemma 2.3 (iv), Var𝑋_𝑛𝑖=𝔼𝑋²_𝑛𝑖−𝔼𝑋_𝑛𝑖²∼𝑙𝜂_𝑛,^𝑛_𝑖=1Var𝑐_𝑛,𝑖𝑋_𝑛𝑖∼10𝑛3𝑙𝜂_𝑛∶=𝐵²_𝑛. (2.8) By (1.5) and Lemma 2.3 (i), max_{1≤𝑖≤𝑛}𝑐_𝑛,𝑖≤2max_{1≤𝑖≤𝑛}𝑏_𝑛,𝑖≤2𝑏_𝑛,1∼2ln𝑛,ln𝑛max_{1≤𝑖≤𝑛}𝔼𝑋_𝑛𝑖𝐵_𝑛⟶0. (2.9) Thus by combining Lemma 2.3 (iv), (1.6), and (2.8), Lindeberg condition 1𝐵²_𝑛^𝑛_𝑖=1𝔼𝑐_𝑛,𝑖𝑋_𝑛𝑖²𝐼_{(|𝑐𝑛,𝑖𝑋𝑛𝑖−𝔼𝑐𝑛,𝑖𝑋𝑛𝑖|>𝜀𝐵𝑛)}∼𝑐𝑛𝑙𝜂_𝑛^𝑛_𝑖=1𝑐²_𝑛,𝑖𝔼𝑋²_𝑛𝑖𝐼_{(|𝑋𝑛𝑖−𝔼𝑋𝑛𝑖|>𝜀𝐵𝑛/2ln𝑛)}≤𝑐𝑛𝑙𝜂_𝑛^𝑛_𝑖=1𝑐²_𝑛,𝑖𝔼𝑋²_𝑛𝑖𝐼_{(|𝑋𝑛𝑖|>𝜀𝐵𝑛/4ln𝑛)}=𝑐𝑛𝑙𝜂_𝑛^𝑛_𝑖=1𝑐²_𝑛,𝑖𝔼𝑋−𝜇²𝐼_{(𝑐𝜂𝑛/ln𝑛<|𝑋−𝜇|≤𝜂𝑛)}𝑙𝜂=𝑐^𝑛−𝑙𝑐𝜂_𝑛/ln𝑛𝑙𝜂_𝑛⟶0,as𝑛⟶∞ (2.10) hold.
Hence, it follows that 𝑆_𝑛,𝑛−𝔼𝑆_𝑛,𝑛𝐵_𝑛^𝑑⟶𝒩,as𝑛⟶∞. (2.11) This implies that for any 𝑔(𝑥), which is a nonnegative, bounded Lipschitz function, 𝔼𝑔𝑆_𝑛,𝑛−𝔼𝑆_𝑛,𝑛𝐵_𝑛⟶𝔼𝑔(𝒩),as𝑛⟶∞. (2.12) Hence, we obtain lim_𝑛→∞1𝐷_𝑛^𝑛_𝑘=1𝑑_𝑘𝔼𝑔𝑆_𝑘,𝑘−𝔼𝑆_𝑘,𝑘𝐵_𝑘=𝔼𝑔(𝒩) (2.13) from the Toeplitz lemma.
On the other hand, note that (2.4) is equivalent to lim_𝑛→∞1𝐷_𝑛^𝑛_𝑘=1𝑑_𝑘𝑔𝑆_𝑘,𝑘−𝔼𝑆_𝑘,𝑘𝐵_𝑘=𝔼𝑔(𝒩)a.s., (2.14) from Theorem  7.1 of Billingsley [17] and Section  2 of Peligrad and Shao [18]. Hence, to prove (2.4), it suffices to prove lim_𝑛→∞1𝐷_𝑛^𝑛_𝑘=1𝑑_𝑘𝑔𝑆_𝑘,𝑘−𝔼𝑆_𝑘,𝑘𝐵_𝑘−𝔼𝑔𝑆_𝑘,𝑘−𝔼𝑆_𝑘,𝑘𝐵_𝑘=0a.s., (2.15) for any 𝑔(𝑥) which is a nonnegative, bounded Lipschitz function.
Let 𝜉_𝑘=𝑔𝑆_𝑘,𝑘−𝔼𝑆_𝑘,𝑘𝐵_𝑘−𝔼𝑔𝑆_𝑘,𝑘−𝔼𝑆_𝑘,𝑘𝐵_𝑘,for𝑘≥1. (2.16)
Clearly, there is a constant 𝑐>0 such that 𝑔(𝑥)≤𝑐,𝑔(𝑥)−𝑔𝑦≤𝑐𝑥−𝑦forany𝑥,𝑦∈ℝ,𝜉_𝑘≤2𝑐,forany𝑘. (2.17)
For any 1≤𝑘<𝑙, note that 𝑆_𝑙,𝑙−𝑆_𝑙,𝑘=^𝑙_𝑖=1𝑐_𝑙,𝑖𝑋_𝑙𝑖−^𝑘_𝑖=1𝑐_𝑙,𝑖𝑋_𝑙𝑖=^𝑙_𝑖=𝑘+1𝑐_𝑙,𝑖𝑋_𝑙𝑖. (2.18)
For any 1≤𝑘<𝑗, note that 𝑔((𝑆_𝑘,𝑘−𝔼𝑆_𝑘,𝑘)/𝐵_𝑘) and 𝑔((𝑆_𝑗,𝑗−𝔼𝑆_𝑗,𝑗−(𝑆_𝑗,𝑘−𝔼𝑆_𝑗,𝑘))/𝐵_𝑗) are independent, 𝑔(𝑥) is a nonnegative, bounded Lipschitz function, ∑^𝑘_𝑖=1𝑐²_𝑘,𝑖∼10𝑘/3,𝑐²_𝑗,𝑖≤4𝑏²_𝑗,𝑖, ∑^𝑘_𝑖=1𝑏²_𝑘,𝑖∼2𝑘, and ln𝑥≤4𝑥^1/4,𝑥≥1. By the definition of 𝜂_𝑗, we get 𝔼𝜉_𝑘𝜉_𝑗=Cov𝑔𝑆_𝑘,𝑘−𝔼𝑆_𝑘,𝑘𝐵_𝑘,𝑔𝑆_𝑗,𝑗−𝔼𝑆_𝑗,𝑗𝐵_𝑗=Cov𝑔𝑆_𝑘,𝑘−𝔼𝑆_𝑘,𝑘𝐵_𝑘,𝑔𝑆_𝑗,𝑗−𝔼𝑆_𝑗,𝑗𝐵_𝑗−𝑔𝑆_𝑗,𝑗−𝔼𝑆_𝑗,𝑗−𝑆_𝑗,𝑘−𝔼𝑆_𝑗,𝑘𝐵_𝑗𝔼∑≤𝑐^𝑘𝑖=1𝑐_𝑗,𝑖𝑋_𝑗𝑖−𝔼𝑋_𝑗𝑖𝑗𝑙𝜂_𝑗≤𝑐𝔼∑^𝑘_𝑖=1𝑐_𝑗,𝑖(𝑋_𝑗𝑖−𝔼𝑋_𝑗𝑖)²𝑗𝑙𝜂_𝑗≤𝑐∑^𝑘_𝑖=1𝑏²_𝑗,𝑖𝔼𝑋²_𝑗𝑖𝑗𝑙𝜂_𝑗≤𝑐∑^𝑘_𝑖=1𝑏_𝑘,𝑖+𝑏_𝑗,𝑘+1²𝑙𝜂_𝑗𝑗𝑙𝜂_𝑗=𝑐∑^𝑘_𝑖=1𝑏²_𝑘,𝑖+∑^𝑘_𝑖=1𝑏²_𝑗,𝑘+1𝑗≤𝑐𝑘+𝑘ln²𝑗/𝑘𝑗𝑘≤𝑐𝑗^1/4. (2.19) By Lemma 2.2, (2.15) holds.
Now we prove (2.5). Let 𝑍_𝑘=𝐼^𝑘_𝑖=1𝑋_𝑖−𝜇>𝜂_𝑘−𝔼𝐼^𝑘_𝑖=1𝑋_𝑖−𝜇>𝜂_𝑘forany𝑘≥1. (2.20) 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 (iii) and (1.5), we get ℙ𝑋−𝜇>𝜂_𝑗(1)𝑙𝜂=𝑜^𝑗𝜂²_𝑗=𝑜(1)𝑗. (2.21) Hence for 1≤𝑘<𝑗, 𝔼𝑍_𝑘𝑍_𝑗=Cov𝐼^𝑘_𝑖=1𝑋_𝑖−𝜇>𝜂_𝑘,𝐼^𝑗_𝑖=1𝑋_𝑖−𝜇>𝜂_𝑗=Cov𝐼^𝑘_𝑖=1𝑋_𝑖−𝜇>𝜂_𝑘,𝐼^𝑗_𝑖=1𝑋_𝑖−𝜇>𝜂_𝑗−𝐼^𝑗_𝑖=𝑘+1𝑋_𝑖−𝜇>𝜂_𝑗≤𝔼𝐼^𝑗_𝑖=1𝑋_𝑖−𝜇>𝜂_𝑗−𝐼^𝑗_𝑖=𝑘+1𝑋_𝑖−𝜇>𝜂_𝑗≤𝔼𝐼^𝑘_𝑖=1𝑋_𝑖−𝜇>𝜂_𝑗≤𝑘ℙ𝑋−𝜇>𝜂_𝑗≤𝑘𝑗. (2.22) By Lemma 2.2, (2.5) holds.
Finally, we prove (2.6). Let 𝜁_𝑘=𝑓𝑉²_𝑘𝑘𝑙𝜂_𝑘−𝔼𝑓𝑉²_𝑘𝑘𝑙𝜂_𝑘forany𝑘≥1. (2.23) For 1≤𝑘<𝑗, 𝔼𝜁_𝑘𝜁_𝑗=Cov𝑓𝑉²_𝑘𝑘𝑙𝜂_𝑘,𝑓𝑉²_𝑗𝑗𝑙𝜂_𝑗=Cov𝑓𝑉²_𝑘𝑘𝑙𝜂_𝑘,𝑓𝑉²_𝑗𝑗𝑙𝜂_𝑗−𝑓𝑉²_𝑗−∑^𝑘_𝑖=1𝑋_𝑖−𝜇²𝐼𝑋_𝑖−𝜇≤𝜂_𝑗𝑗𝑙𝜂_𝑗𝔼∑≤𝑐^𝑘𝑖=1𝑋_𝑖−𝜇²𝐼𝑋_𝑖−𝜇≤𝜂_𝑗𝑗𝑙𝜂_𝑗=𝑐𝑘𝔼𝑋−𝜇²𝐼𝑋−𝜇≤𝜂_𝑗𝑗𝑙𝜂_𝑗𝜂=𝑐𝑘𝑙_𝑗𝑗𝑙𝜂_𝑗𝑘=𝑐𝑗. (2.24) By Lemma 2.2, (2.6) holds. This completes the proof of Lemma 2.4.

Proof of Theorem 1.1. Let 𝑍_𝑗=𝑇_𝑗/(𝑗(𝑗+1)𝜇/2); then (1.8) is equivalent to lim_𝑛→∞1𝐷_𝑛^𝑛_𝑘=1𝑑_𝑘𝐼√3𝜇√10𝑉_𝑘^𝑘_𝑗=1ln𝑍_𝑗≤𝑥=Φ(𝑥),a.s.forany𝑥, (2.25) where Φ(𝑥) is the distribution function of the standard normal random variable 𝒩.
Let 𝑞∈(4/3,2), then 𝔼|𝑋|^𝑞<∞. Using Marcinkiewicz-Zygmund strong large number law, we have 𝑆_𝑘−𝜇𝑘=𝑜𝑘^1/𝑞a.s. (2.26) Thus, 𝑍_𝑖=∑−1^𝑖𝑗=1𝑆_𝑗−𝑖(𝑖+1)𝜇/2𝑖()≤∑𝑖+1𝜇/2^𝑖𝑗=1𝑆_𝑗−𝜇𝑗𝑖()≤∑𝑖+1𝜇/2^𝑖𝑗=1𝑗^1/𝑞𝑖()𝑖𝑖+1𝜇/2≤𝑐^1/𝑞+1𝑖²=𝑖^1/𝑞−1⟶0,a.s. (2.27) Hence by |ln(1+𝑥)−𝑥|=𝑂(𝑥²) for |𝑥|<1/2, for any 0<𝜀<1, 1√1±𝜀𝐵_𝑘^𝑘_𝑖=1ln𝑍_𝑖−1√1±𝜀𝐵_𝑘^𝑘_𝑖=1𝑍_𝑖1−1≤𝑐𝑘𝑙𝜂_𝑘^𝑘𝑖=1𝑍_𝑖−1²≤𝑐𝑘𝑙𝜂_𝑘^𝑘𝑖=1𝑖^{2(1/𝑞−1)}1≤𝑐𝑘^3/2−2/𝑞𝑙𝜂_𝑘⟶0a.s.𝑘⟶∞, (2.28) from 3/2−2/𝑞>0, 𝑙(𝑥) is a slowly varying function at ∞, and 𝜂_𝑘≤𝑘+1.
Hence for almost every event 𝜔 and any 𝛿>0, there exists 𝑘₀=𝑘₀(𝜔,𝛿,𝑥) such that for 𝑘>𝑘₀, 𝐼𝜇√1±𝜀𝐵_𝑘^𝑘_𝑖=1𝑍_𝑖𝜇−1≤𝑥−𝛿≤𝐼√1±𝜀𝐵_𝑘^𝑘_𝑖=1ln𝑍_𝑖𝜇≤𝑥≤𝐼√1±𝜀𝐵_𝑘^𝑘_𝑖=1𝑍_𝑖−1≤𝑥+𝛿. (2.29)
Note that under condition |𝑋_𝑗−𝜇|≤𝜂_𝑘,1≤𝑗≤𝑘, 𝜇^𝑘_𝑖=1𝑍_𝑖−1=^𝑘_𝑖=1∑^𝑖_𝑙=1𝑆_𝑙−𝜇∑^𝑖_𝑙=1𝑙𝑖()=𝑖+1/2^𝑘_𝑖=11𝑖(𝑖+1)/2^𝑖𝑙_𝑙=1𝑗=1𝑋_𝑗=−𝜇_𝑘𝑖=11𝑖(𝑖+1)/2^𝑖𝑖_{𝑗=1𝑙=𝑗}𝑋_𝑗−𝜇=^𝑘𝑖_𝑖=1𝑗=12𝑖+1−𝑗𝑖()𝑋𝑖+1^𝑗=−𝜇_{𝑘𝑘𝑗=1𝑖=𝑗}2𝑖+1−𝑗𝑖(𝑖+1)𝑋_𝑘𝑗=^𝑘_𝑗=1𝑐_𝑘,𝑗𝑋_𝑘𝑗=𝑆_𝑘,𝑘. (2.30)
Thus, for any given 0<𝜀<1, 𝛿>0, combining (2.29), we have for 𝑘>𝑘₀𝐼√3𝜇∑^𝑘_𝑖=1ln𝑍_𝑖√10𝑉_𝑘√≤𝑥≤𝐼3𝜇∑^𝑘_𝑖=1ln𝑍_𝑖10(1+𝜀)𝑘𝑙𝜂_𝑘≤𝑥+𝐼𝑉²_𝑘>(1+𝜀)𝑘𝑙𝜂_𝑘+𝐼^𝑘_𝑖=1𝑋_𝑖−𝜇>𝜂_𝑘≤𝐼𝜇∑^𝑘_𝑖=1𝑍_𝑖−1√1+𝜀𝐵_𝑘≤𝑥+𝛿+𝐼𝑉²_𝑘>(1+𝜀)𝑘𝑙𝜂_𝑘+𝐼^𝑘_𝑖=1𝑋_𝑖−𝜇>𝜂_𝑘≤𝐼𝑆_𝑘,𝑘√1+𝜀𝐵_𝑘≤𝑥+𝛿+𝐼𝑉²_𝑘>(1+𝜀)𝑘𝑙𝜂_𝑘+2𝐼^𝑘_𝑖=1𝑋_𝑖−𝜇>𝜂_𝑘𝐼√for𝑥≥0,3𝜇∑^𝑘_𝑖=1ln𝑍_𝑖√10𝑉_𝑘≤𝑥≤𝐼𝑆_𝑘,𝑘√1−𝜀𝐵_𝑘≤𝑥+𝛿+𝐼𝑉²_𝑘<(1−𝜀)𝑘𝑙𝜂_𝑘+2𝐼^𝑘_𝑖=1𝑋_𝑖−𝜇>𝜂_𝑘𝐼√for𝑥<0,3𝜇∑^𝑘_𝑖=1ln𝑍_𝑖√10𝑉_𝑘≤𝑥≥𝐼𝑆_𝑘,𝑘√1−𝜀𝐵_𝑘≤𝑥−𝛿−𝐼𝑉²_𝑘<(1−𝜀)𝑘𝑙𝜂_𝑘−2𝐼^𝑘_𝑖=1𝑋_𝑖−𝜇>𝜂_𝑘𝐼√,for𝑥≥0,3𝜇∑^𝑘_𝑖=1ln𝑍_𝑖√10𝑉_𝑘≤𝑥≥𝐼𝑆_𝑘,𝑘√1+𝜀𝐵_𝑘≤𝑥−𝛿−𝐼𝑉²_𝑘>(1+𝜀)𝑘𝑙𝜂_𝑘−2𝐼^𝑘_𝑖=1𝑋_𝑖−𝜇>𝜂_𝑘,for𝑥<0. (2.31) Therefore, to prove (2.25), it suffices to prove lim_𝑛→∞1𝐷_𝑛^𝑛_𝑘=1𝑑_𝑘𝐼𝑆_𝑘,𝑘𝐵_𝑘≤√1±𝜀𝑥±𝛿₁=Φ√1±𝜀𝑥±𝛿₁a.s., (2.32)lim_𝑛→∞1𝐷_𝑛^𝑛_𝑘=1𝑑_𝑘𝐼^𝑘_𝑖=1𝑋_𝑖−𝜇>𝜂_𝑘=0a.s., (2.33)lim_𝑛→∞1𝐷_𝑛^𝑛_𝑘=1𝑑_𝑘𝐼𝑉²_𝑘>(1+𝜀)𝑘𝑙𝜂_𝑘=0a.s., (2.34)lim_𝑛→∞1𝐷_𝑛^𝑛_𝑘=1𝑑_𝑘𝐼𝑉²_𝑘<(1−𝜀)𝑘𝑙𝜂_𝑘=0a.s, (2.35) for any 0<𝜀<1 and 𝛿₁>0.
Firstly, we prove (2.32). Let 0<𝛽<1/2 and ℎ(⋅) be a real function, such that for any given 𝑥∈ℝ, 𝐼𝑦≤√1±𝜀𝑥±𝛿₁−𝛽≤ℎ𝑦≤𝐼𝑦≤√1±𝜀𝑥±𝛿₁+𝛽. (2.36)
By 𝔼(𝑋_𝑖−𝜇)=0, Lemma 2.1 (iv) and (1.5), we have 𝔼𝑆_𝑘,𝑘=𝔼^𝑘_𝑖=1𝑐_𝑘,𝑖𝑋_𝑖−𝜇𝐼𝑋_𝑖−𝜇≤𝜂_𝑘≤2^𝑘_𝑖=1𝑏_𝑘,𝑖𝔼𝑋_𝑖−𝜇𝐼𝑋_𝑖−𝜇>𝜂_𝑘=2^𝑘𝑘_{𝑖=1𝑗=𝑖}1𝑗𝔼𝑋−𝜇𝐼𝑋−𝜇>𝜂_𝑘=^𝑘𝑗_𝑗=1𝑖=11𝑗𝑜𝑙𝜂_𝑘𝜂_𝑘=𝑜𝑘𝑙𝜂_𝑘. (2.37) This, combining with (2.4), (2.36) and the arbitrariness of 𝛽 in (2.36), (2.32) holds.
By (2.5), (2.21) and the Toeplitz lemma, 10≤𝐷_𝑛^𝑛_𝑘=1𝑑_𝑘𝐼^𝑘_𝑖=1𝑋_𝑖−𝜇>𝜂_𝑘∼1𝐷_𝑛^𝑛_𝑘=1𝑑_𝑘𝔼𝐼^𝑘_𝑖=1𝑋_𝑖−𝜇>𝜂_𝑘≤1𝐷_𝑛^𝑛_𝑘=1𝑑_𝑘𝑘ℙ𝑋−𝜇>𝜂_𝑘⟶0a.s. (2.38) That is, (2.33) holds.
Now we prove (2.34). For any given 𝜀>0, let 𝑓 be a nonnegative, bounded Lipschitz function such that 𝐼()(𝑥)𝜀𝑥>1+𝜀≤𝑓≤𝐼𝑥>1+2. (2.39)
From 𝔼𝑉²_𝑘=𝑘𝑙(𝜂_𝑘), 𝑋_𝑘𝑖 is i.i.d., 𝔼(𝑋²_𝑘𝑖−𝔼𝑋²_𝑘𝑖)=0, Lemma 2.1 (v), and (1.5), ℙ𝑉²_𝑘>𝜀1+2𝑘𝑙𝜂_𝑘=ℙ𝑉²_𝑘−𝔼𝑉²_𝑘>𝜀2𝑘𝑙𝜂_𝑘𝔼≤𝑐𝑉²_𝑘−𝔼𝑉²_𝑘²𝑘²𝑙²𝜂_𝑘𝔼≤𝑐𝑋−𝜇⁴𝐼𝑋−𝜇≤𝜂_𝑘𝑘𝑙²𝜂_𝑘=𝑜(1)𝜂²_𝑘𝑘𝑙𝜂_𝑘=𝑜(1)⟶0. (2.40) Therefore, combining (2.6) and the Toeplitz lemma, 10≤𝐷_𝑛^𝑛_𝑘=1𝑑_𝑘𝐼𝑉²_𝑘>(1+𝜀)𝑘𝑙𝜂_𝑘≤1𝐷_𝑛^𝑛_𝑘=1𝑑_𝑘𝑓𝑉²_𝑘𝑘𝑙𝜂_𝑘∼1𝐷_𝑛^𝑛_𝑘=1𝑑_𝑘𝔼𝑓𝑉²_𝑘𝑘𝑙𝜂_𝑘≤1𝐷_𝑛^𝑛_𝑘=1𝑑_𝑘𝔼𝐼𝑉²_𝑘>𝜀1+2𝑘𝑙𝜂_𝑘=1𝐷_𝑛^𝑛_𝑘=1𝑑_𝑘ℙ𝑉²_𝑘>𝜀1+2𝑘𝑙𝜂_𝑘⟶0a.s. (2.41) Hence, (2.34) holds. By similar methods used to prove (2.34), we can prove (2.35). This completes the proof of Theorem 1.1.

Acknowledgments

The author is 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 paper is supported by the National Natural Science Foundation of China (11061012), a 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).