Cramér Type Moderate deviations for the Maximum of Self-normalized Sums

Let $\{ X, X_i , i \geq 1\}$ be i.i.d. random variables, $S_k$ be the partial sum and $V_n^2 = \sum_{1\leq i\leq n} X_i^2$. Assume that $E(X)=0$ and $E(X^4) < \infty$. In this paper we discuss the moderate deviations of the maximum of the self-normalized sums. In particular, we prove that $P(\max_{1 \leq k \leq n} S_k \geq x V_n) / (1- \Phi(x)) \to 2$ uniformly in $x \in [0, o(n^{1/6}))$.


Introduction and main results
Let X , X 1 , X 2 , • • • be a sequence of i.i.d.random variables with mean zero.Set The past decade has witnessed a significant development on the limit theorems for the so-called self-normalized sum S n /V n .Griffin and Kuelbs (1989) obtained a self-normalized law of the iterated logarithm for all distributions in the domain of attraction of a normal or stable law.Shao (1997) showed that no moment conditions are needed for a self-normalized large deviation result P(S n /V n ≥ x n) and that the tail probability of S n /V n is Gaussian like when X 1 is in the domain of attraction of the normal law and sub-Gaussian like when X is in the domain of attraction of a stable law, while Giné, Götze and Mason (1997) proved that the tails of S n /V n are uniformly sub-Gaussian when the sequence is stochastically bounded.Shao (1999) established a Cramér type result for selfnormalized sums only under a finite third moment condition.Jing, Shao and Wang (2003) proved a Cramér type large deviation result (for independent random variables) under a Lindeberg type condition.Jing, Shao and Zhou (2004) obtained the saddlepoint approximation without any moment condition.Other results include Wang and Jing (1999) as well as Robinson and Wang (2005) for an exponential non-uniform Berry-Esseen bound, Csörgő, Szyszkowicz and Wang (2003a, b) for Darling-Erdős theorems and Donsker's theorems, Wang (2005) for a refined moderate deviation, Hall and Wang (2004) for exact convergence rates, and Chistyakov and Götze (2004) for all possible limiting distributions when X is in the domain of attraction of a stable law.These results show that the self-normalized limit theorems usually require fewer moment conditions than the classical limit theorems do.On the other hand, self-normalization is commonly used in statistics.Many statistical inferences require the use of classical limit theorems.However, these classical results often involve some unknown parameters, one needs to first estimate the unknown parameters and then substitute the estimators into the classical limit theorems.This commonly used practice is exactly the self-normalization.Hence, the development on self-normalized limit theorems not only provides a theoretical foundation for statistical practice but also gives a much wider applicability of the results because they usually require much less moment assumptions.
In contrast with the achievements for the self-normalized partial sum S n /V n , there is little work on the maximum of self-normalized sums.This paper is part of our efforts to develop limit theorems for the maximum of self-normalized sums.Using a different approach from some known techniques for self-normalized sum, we establish a Cramér type large deviation result for the maximum of selfnormalized sums under a finite fourth moment.Note that the Cramér type large deviation result holds under a finite third moment for partial sum, we conjecture that a finite third moment is sufficient for (1.1).Our main result is as follows.
This theorem is comparable to the large deviation result for the maximum of partial sum given in Aleshkyavichene (1979).However the latter requires a finite exponential moment condition.We also note that Berry-Esseen type results for the maximum of non-self-normalized sums are available in literature and one can refer to Arak (1974) and Arak and Nevzorov (1973).For a Chernoff type large deviation, the fourth moment condition can be relaxed to a finite second moment condition.Indeed, we have the following theorem.
Theorem 1.2.If X is in the domain of attraction of the normal law, then This paper is organized as follows.In the next section, we give the proof of Theorem 1.1 as well as two propositions.The proofs of the two propositions are postponed to Section 3. Finally, the proof of Theorem 1.2 is given in Section 4.

Proof of Theorem 1.1
Throughout this section, without loss of generality, we assume E(X 2 ) = 1.The proof is based on the following two propositions.Their proofs will be given in the next section.
where C is a constant not depending on x and n.
), where ε n → 0 is any sequence of constants.
We are now ready to prove Theorem 1.1.Let M n = max 1≤k≤n S k .It is well-known that if the second moment of X is finite, then by the law of large numbers and the weak convergence Hence (1.1) holds for x ∈ [0, 2] and we can assume 2 ≤ x = o(n 1/6 ).We first prove 2).It is readily seen by the independence of X j and max 1≤k≤n S n , the iid properties of X i and Proposition 2.1 that if EX 4 < ∞, then for each j uniformly in x ∈ [2, o(n 1/6 )).This, together with Proposition 2.2 yields )).This proves (2.4).
We next prove where S(j) k and V ( j) n are defined similarly as n with Xi to replace X i .By (2.7), result (2.6) follows immediately from Proposition 2.2.The proof of Theorem 1.1 is now complete.

Preliminary lemmas
This subsection provides several preliminary lemmas.Some of which are interesting by themselves.For each n ≥ 1, let X n,i , 1 ≤ i ≤ n, be a sequence of independent random variables with zero mean and finite variance.Write , and there exists an R 0 (that may depend on x and n) ) , where 0 < ε n → 0 is any sequence of constants.Furthermore we also have Proof.The result (3.1) follows immediately from ( 4) and ( 5) of Sakhanenko (1991).In order to prove (3.2), without loss of generality, assume x ≥ 1.The result for 0 ≤ x ≤ 1 follows from the well-known Berry-Esseen bound.See Nevzorov (1973), for instance.For each ε > 0, write, for k = 1, 2, ..., n, S and where c 0 > 0 is an absolute constant.
The statement (3.3) has been established in (8) of Nevzorov (1973).We present a proof here for the convenience of the reader.Let S (3) We first claim that there exists an absolute constant c 0 > 0 such that, for any n ≥ 1, 1 ≤ l ≤ n and ε > 0, ) In fact, by letting , it follows from the non-uniform Berry-Esseen bound that, for any n ≥ 1, 1 ≤ l ≤ n and ε > 0, where A 0 is an absolute constant and t 0 = c 0 ε/s n .Note that t 0 e −s 2 /2 ds ≥ t(1 + t) −3 /2 for any t ≥ 0. Simple calculations yield (3.4), by choosing c 0 ≥ 4A 0 2π.The proof of (3.5) is similar, we omit the details.Now, by noting and hence for y = x B n , Similarly, it follows from X n,i I (X n,i ≤ε) ≤ X n,i I (|X n,i |≤ε) and X n,i I (X n,i ≤ε) ≤ X n,i that, for all 1 ≤ l ≤ n, P{S (1)  n,n − S (1) and hence for y = x B n , n,k ≥ y} (1)  n,n ≥ y + (1 + c 0 )ε + A n }.This completes the proof of (3.3).
In the following proof, take ε = ε n B n /x in (3.3).By recalling EX n,i = 0, we have where . Therefore, for n large enough such that ε n ≤ 1/4, P S (2)  n,n where . Similarly, we have where . By virtue of (3.3), (3.7)-(3.8)and the well-known fact that if uniformly in x ∈ [1, ∞), the result (3.2) will follow if we prove . In fact, by noting var( and n sufficient large, (3.10) follows immediately from (3.1).The proof of Lemma 3.1 is now complete.Lemma 3.1 will be used to establish Cra ḿer type large deviation result for truncated random variables under finite moment conditions.Indeed, as a consequence of Lemma 3.1, we have the following lemma.Lemma 3.2.If EX 2 = 1 and E|X | (α+2)/α < ∞, 0 < α ≤ 1, then we have and uniformly in the range 2 ≤ x ≤ ε n n 1/6 , where 0 < ε n → 0 is any sequence of constants, and X j and Sk are defined as in (2.2). Proof.
, where C 0 is a constant not depending on x and n.So, by (3.1) in Lemma 3.1, uniformly in the range 2 ≤ x ≤ ε n n 1/6 .Now, by noting it follows from (3.13) and then (3.9) that uniformly in the range 1 ≤ x ≤ ε n n 1/6 .This proves (3.11).The proof of (3.12) is similar except we make use of (3.2) in replacement of (3.1) and hence the details are omitted.The proof of Lemma 3.2 is now complete.

Also write γn
This, together with f (h) ≤ f * (h), implies that, for all z satisfying |z| < min{1, By virtue of (3.17) and the identity Now, by noting that it follows easily from (3.18) and the i.i.d.properties of X j , j ≥ 1 that This, together with the facts that γ0 (h) = 1 and gives (3.15).
We next prove (3.16).In view of (3.14) and (3.15), it is enough to show that where C is a constant not depending on x and n.In order to prove (3.19), let ε 0 be a constant such that 0 < ε 0 < min{1, λ 2 /(6θ )}, and t 0 > 0 sufficiently small such that First note that It follows from the inequality e x − 1 − x ≤ |x| 3/2 e x∨0 and the iid properties of X j that As in the proof of Lemma 1 of Aleshkyavichene(1979) [see (26) and ( 28) there], we have for all k ≥ 1 and sufficient small b, where and C is a constant not depending on x and n.Taking these estimates into (3.20),we obtain for sufficient small b.On the other hand, it follows easily from (3.14) that there exists a sufficient small b 0 such that for all 0 and also f (b) ≥ e −ǫ 1 /2 .Now, by recalling ε 0 < λ 2 /(6θ ) and using the relation where ρ k = O(k −3/2 ), simple calculations show that there exists a sufficient small b 0 such that for all 0 where C 1 and C 2 are constants not depending on x and n.This proves (3.19) and hence completes the proof of (3.16).

Proofs of propositions.
Without loss of generality, assume EX 2 = 1.Otherwise we only need to replace X j by X j / EX 2 .
Proof of Proposition 2.1.Since (2.1) is trivial for 0 ≤ x < 1, we assume x ≥ 1. Write b = x/ n.Observe that (3.21) By (3.16) with λ = 1 and θ = 1/2 in Lemma 3.3 and the exponential inequality, we have whenever 0 ≤ x ≤ n 1/6 , where C is a constant not depending on x and n.By (3.16) again, it follows from the similar arguments as in the proofs of (2.12) and (2.28) in Shao (1999) that and E|X | max{(α+2)/α,4} < ∞ that (by letting t = x/ n) uniformly in the range 2 ≤ x ≤ ε n n 1/6 .It is now readily seen that uniformly in the range 2 ≤ x ≤ ε n n 1/6 .Therefore, in order to prove Proposition 2.2, it suffices to show that In fact, by noting x Vn ≤ (x 2 + b 2 V 2 n )/2b where b = x/ n, we have This, together with (3.23), implies that (3.24) will follow if we prove uniformly in the range 2 ≤ x ≤ ε n n 1/6 .
In order to prove (3.25), write ξ = and we have where we have used the fact that, for sufficient small b, Now, for any 0 < θ 0 ≤ ( n/x) 1−α and b = x/ n, we have uniformly in the range 1 ≤ x ≤ ε n n 1/6 , where we have used the fact that Then z n → ∞ and nl(z n ) = x 2 n z 2 n for n large enough.For any 0 < ǫ < 1/2, we have where To see this, it sufficies to note that max 1≤k≤n As in Shao(1997), we have As for J 1 , we have Then by a Lévy inequality, we have where the last equality is from Shao(1997).Now (4.1) follows from the above inequalities and the arbitrariness of ǫ.The proof of Theorem 1.1 is complete.
.27) Therefore, by a similar argument as in the proof of (3.15) in Lemma 3.3 and noting φk (b) :