Limit Theorems for Self-normalized Large Deviation

Let X, X 1 , X 2 , · · · be i.i.d. random variables with zero mean and £nite variance σ 2. It is well known that a £nite exponential moment assumption is necessary to study limit theorems for large deviation for the standardized partial sums. In this paper, limit theorems for large deviation for self-normalized sums are derived only under £nite moment conditions. In particular , we show that, if EX 4 < ∞, then


Introduction and main results
Let X, X 1 , X 2 , • • • , be a sequence of non-degenerate independent and identically distributed (i.i.d.) random variables with zero mean.Set The self-normalized version of the classical central limit theorem states that, as n → ∞, if and only if the distribution of X is in the domain of attraction of the normal law, where Φ(x) denotes the standard normal distribution function.This beautiful self-normalized central limit theorem was conjectured by Logan, Mallows, Rice and Shepp (1973), and latterly proved by Gine, Götze and Mason (1997).For a short summary of developments that have eventually led to Gine, Götze and Mason (1997), we refer to the Introduction of the latter paper.
Result (1.2) is useful in statistics because it provides not only the relative error but also a Berry-Esseen type rate of convergence.Indeed, as a direct consequence of this result, it has been shown in Jing, Shao and Wang (2003) that bootstrapped studentized t-statistics possess large deviation properties in the region 0 ≤ x ≤ o(n 1/6 ) under only a £nite third moment condition.However, (1.1) as well as (1.2) does not capture the term with n −1/2 explicitly.This short has limited further applications of the self-normaized large deviation.
In this paper we investigate the limit theorems for self-normalized large deviation.Under £nite moment conditions, a leading term with n −1/2 in (1.1) and (1.2) is obtained explicitly.
THEOREM 1.2.Assume that E|X| 3 < ∞.Then, there exists a positive absolute constant c where A is an absolute positive constant.
REMARK 1.1.Similar results to those in Theorem 1.1 hold for the standardized mean under much stronger conditions.For instance, it follows from Section 5.8 of Petrov (1995) that, for x ≥ 0 and x = O(n 1/6 ), hold only when Cramér's condition is satis£ed, i.e., Ee tX < ∞ for t being in a neighborhood of zero.We also notice that there are different formulae for the self-normalized and standardized cases.
Hence − x 3 EX 3 3 √ nσ 3 in (1.6) provides a leading term in this case.However it remains an open problem for more re£ned results.
This paper is organized as follows.The proofs of main results will be given in Section 3.
Next section we present two auxiliary theorems that will be used in the proofs of main results.
The proofs of these auxiliary theorems will be postponed to Sections 4 and 5 respectively.
Without loss of generality, throughout the paper, we assume σ 2 = EX 2 = 1 and denote by • • • absolute positive constants, which may be different at each occurrence.If a constant depends on a parameter, say u, then we write A(u).In addition to the notation for L n,x and ∆ n,x de£ned in Theorem 1.2, we always let

Two auxiliary theorems
Throughout the section we assume that X, X 1 , X 2 , • • • , are i.i.d.random variables satisfying EX = 0, EX 2 = 1 and E|X| 3 < ∞.Two theorems in this section are established under quite general setting, which will be interesting in themselves.The proofs of these two theorems will be given in Sections 4 and 5 respectively.
Proof of Theorem 1.2.Without loss of generality, assume x ≥ 2. If 0 ≤ x < 2, the results are direct consequences of the Berry-Esseen bound (cf.Bentkus and G ötze (1996)) We £rst provide four lemmas.For simplicity of presentation, de£ne τ = √ n/(1 + x) and with c suf£ciently small throughout the section except where we point out.
The proof of (3.1) is similar by using (2.7).We omit the details.The proof of Lemma 3.1 is now complete.
The next lemma is from Lemma 6.4 in Jing, Shao and Wang (2003).
LEMMA 3.4.We have Proof.As in Wang and Jing (1999), for x ≥ 2, (3.9) It follows easily from Lemma 3.3 that for all i This, together with In view of (3.9) and (3.10), the inequalities (3.7) and (3.8) will follow if we prove By the inequality (1 + y) 1/2 ≥ 1 + y/2 − y 2 for any y ≥ −1 and Lemma 3.2, where, after some algebra (see, Jing, Shao and Wang (2003), for example), As in the proof of Lemma 3.1, tedious but elementary calculations show that the inequalities (2.1)-(2.5)hold true for B 2 n = n i=1 E X2 i and the η i de£ned above.Therefore it follows from (3.2) in Theorem 2.1 that for 2 ≤ x ≤ cρ −1 n with c suf£ciently small.Take this estimate back into (3.13),we get the desired (3.11).
After these preliminaries, we are now ready to prove Theorem 1.2.
As is well-known (see Wang and Jing (1999) for example), The left hand inequality of (1.6) follows from Lemma 3.1 immediately.

Proof of Theorem 2.1
The proof of Theorem 2.1 is based on the conjugate method.To employ the method, let .

Proof of Theorem 2.2
The idea for the proof of Theorem 2.2 is similar to Proposition 5.4 in Jing, Shao and Wang (2002), but we need some different details.Throughout this section, we use the following notations: g(t, x) = Ee itζ 1 / √ n and The proof of Theorem 2.2 is based on the following lemmas.

Now, by using |e
where This implies (5.3).The proof of Lemma 5.1 is now complete.
To prove (5.13) and (5.14), put By (2.12) and (5.15) Therefore, (5.13) follows easily from (5.12) and (5.15) with m = n.In view of independence of ζ j , on the other hand, (5.15) implies that for any where we have used the estimate (recalling (2.12)): This gives (5.14).The proof of Lemma 5.2 is now complete.
LEMMA 5.3.Let F be a distribution function with the characteristic function f .Then for all y ∈ R and T > 0 it holds that (5.16) Proof of Lemma 5.3 can be found in Prawitz (1972).
LEMMA 5.5.The integral satisfy that, for any y ∈ R and 0 (5.22)where K 2 (s) is de£ned as in Lemma 5.3 and Proof.We only prove (5.21).Similar to the proof of Lemma 5.4, we assume ρ n ≤ 12 −3 , which 2 e −t 2 /2 .The J + can be rewrite as where J 11 = i π V.P.
By using (5.3) and (5.13), we have Noting that |K 2 (s) − 1| ≤ As 2 , for |s| ≤ 1/2 (cf., e.g., Lemma 2.1 in Bentkus (1994), similar to (5.20), it can be easily shown that On the other hand, simple calculation shows that Therefore, it follows from all these estimates and x ≤ cρ −1 n that This also completes the proof of Lemma 5.5.
After these lemmas, we are now ready to prove Theorem 2.2.