Preliminary lemmas getdocbe1d. 167KB Jun 04 2011 12:04:55 AM

properties of X i that P max 1 ≤k≤n ¯ S k ≥ x ¯ V n ≤ P M n ≥ x V n + n X j=1 P max 1 ≤k≤n ¯ S k ≥ x ¯ V n , X j 6= ¯ X j ≤ P max 1 ≤k≤n S k ≥ x V n + n X j=1 P |X j | p n x 3 4 P max 1 ≤k≤n ¯ S j k ≥ p x 2 − 1 ¯ V j n ≤ P max 1 ≤k≤n S k ≥ x V n +o1x 3 n −12 P max 1 ≤k≤n−1 ¯ S k ≥ p x 2 − 1 ¯ V n −1 , 2.7 where ¯ S j k and ¯ V j n are defined similarly as S j k and V j n with ¯ X i 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. ƒ 3 Proofs of Propositions

3.1 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 S ∗ n,k = P k i=1 X n,i , k = 1, 2, ...n, B 2 n = P n i=1 EX 2 n,i and Lt = n X i=1 E |X n,i | 3 max {e t X n,i , 1 } . Lemma 3.1. If L 3n := P n i=1 E |X n,i | 3 B 3 n → 0, and there exists an R that may depend on x and n such that R ≥ 2 max{x, 1}B n and LR ≤ C P n i=1 E |X n,i | 3 , where C ≥ 1 is a constant, then lim n →∞ PS ∗ n,n ≥ x B n 1 − Φx = 1, 3.1 uniformly in 0 ≤ x ≤ ε n L −13 3n , where 0 ε n → 0 is any sequence of constants. Furthermore we also have lim n →∞ Pmax 1 ≤k≤n S ∗ n,k ≥ x B n 1 − Φx = 2, 3.2 uniformly in 0 ≤ x ≤ ε n L −13 3n . 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 1185 well-known Berry-Esseen bound. See Nevzorov 1973, for instance. For each ε 0, write, for k = 1, 2, ..., n, S 1 n,k = k X i=1 X n,i I X n,i ≤ε , S 2 n,k = k X i=1 X n,i I X n,i ≥−ε and A n = P n i=1 |EX n,i I X n,i ≥−ε | + |EX n,i I X n,i ≤ε | . We have 2P S 1 n,n ≥ x B n + c + 1ε + A n ≤ P max 1 ≤k≤n S ∗ n,k ≥ x B n ≤ 2P S 2 n,n ≥ x B n − c ε − A n , 3.3 where c 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 n,k = P k i=1 X n,i I |X n,i |≤ε for k = 1, 2, ..., n. We first claim that there exists an absolute constant c 0 such that, for any n ≥ 1, 1 ≤ l ≤ n and ε 0, I 1n ≡ P{S 3 n,n − S 3 n,l − ES 3 n,n − S 3 n,l ≥ c ε} ≤ 12, 3.4 I 2n ≡ P{S 3 n,n − S 3 n,l − ES 3 n,n − S 3 n,l ≥ −c ε} ≥ 12. 3.5 In fact, by letting s 2 n = varS 3 n,n − S 3 n,l and Y i = X n,i I |X n,i |≤ε − EX n,i I |X n,i |≤ε , it follows from the non-uniform Berry-Esseen bound that, for any n ≥ 1, 1 ≤ l ≤ n and ε 0, I 1n ≤ 1 − Φc εs n + A 1 + c εs n −3 n X j=l+1 E |Y j | 3 s 3 n ≤ 1 2 − 1 p 2 π Z t e −s 2 2 ds + 2A c 1 + t −3 t , where A is an absolute constant and t = c εs n . Note that R t e −s 2 2 ds ≥ t1 + t −3 2 for any t ≥ 0. Simple calculations yield 3.4, by choosing c ≥ 4A p 2 π. The proof of 3.5 is similar, we omit the details. Now, by noting X n,i I |X n,i |≤ε ≤ X n,i I X n,i ≥−ε and X n,i ≤ X n,i I X n,i ≥−ε , we obtain, for all 1 ≤ l ≤ n, P {S 2 n,n − S 2 n,l ≥ −c ε − A n } ≥ P{S 3 n,n − S 3 n,l − ES 3 n,n − S 3 n,l ≥ −c ε} ≥ 1 2 , and hence for y = x B n , P { max 1 ≤k≤n S ∗ n,k ≥ y} ≤ P{ max 1 ≤k≤n S 2 n,k ≥ y} = n X k=1 P {S 2 n,1 y, · · · , S 2 n,k ≥ y} ≤ 2 n X k=1 P {S 2 n,1 y, · · · , S 2 n,k ≥ y} × P{S 2 n,n − S 2 n,k ≥ −c ε − A n } ≤ 2 n X k=1 P {S 2 n,1 y, · · · , S 2 n,k ≥ y, S 2 n,n ≥ y − c ε − A n } ≤ 2 P{S 2 n,n ≥ y − c ε − A n }. 1186 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 n,l ≥ c ε + A n } ≤ P{S 3 n,n − S 3 n,l − ES 3 n,n − S 3 n,l ≥ c ε} ≤ 1 2 . and hence for y = x B n , P { max 1 ≤k≤n S ∗ n,k ≥ y} ≥ P{ max 1 ≤k≤n S 1 n,k ≥ y} = n X k=1 P {S 1 n,1 y, · · · , S 1 n,k −1 y, S 1 n,k ≥ y} = n X k=1 P {S 1 n,1 y, · · · , S 1 n,k −1 y, y ≤ S 1 n,k ≤ y + ε} ≥ 2 n X k=1 P {S 1 n,1 x, · · · , S 1 n,k −1 y, y ≤ S 1 n,k ≤ y + ε} ×P{S 1 n,n − S 1 n,k ≥ c ε + A n } ≥ 2 n X k=1 P {S 1 n,1 y, · · · , S 1 n,k −1 y, y ≤ S 1 n,k ≤ y + ε, S 1 n,n ≥ y + 1 + c ε + A n } = 2 P {S 1 n,n ≥ y + 1 + c 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 |ES t n,n | ≤ A n ≤ ε −2 P n i=1 E |X n,i | 3 ≤ ε n B n x and varS t n,n = B 2 n + rn, t = 1, 2, 3.6 where |rn| ≤ 2ε −1 P n i=1 E |X n,i | 3 ≤ 2ε 2 n B 2 n x 2 , whenever 1 ≤ x ≤ ε n L −13 3n . Therefore, for n large enough such that ε n ≤ 14, P S 2 n,n ≥ x B n − c ε − A n ≤ P S 2 n,n − ES 2 n,n ≥ x B n [1 − c + 1ε n x 2 ] = P S 2 n,n − ES 2 n,n ≥ x Æ varS 2 n,n 1 + η 2n , 3.7 where |η 2n | = | Ç B 2 n varS 2 n,n [1 − c + 1ε n x 2 ] − 1| ≤ 2c + 2ε n x 2 by 3.6, uniformly in 1 ≤ x ≤ ε n L −13 3n . Similarly, we have P S 1 n,n ≥ x B n + c + 1ε + A n ≥ P S 1 n,n − ES 1 n,n ≥ x Æ varS 1 n,n 1 + η 1n , 3.8 where |η 1n | ≤ 2c +3ε n x 2 by 3.6, uniformly in 1 ≤ x ≤ ε n L −13 3n . By virtue of 3.3, 3.7-3.8 and the well-known fact that if |δ n | ≤ Cε n x 2 , then lim n →∞ 1 − Φ x1 + δ n 1 − Φx = 1, 3.9 1187 uniformly in x ∈ [1, ∞, the result 3.2 will follow if we prove lim n →∞ P S t n,n − ES t n,n ≥ x Æ varS t n,n 1 − Φx = 1, for t = 1, 2, 3.10 uniformly in 0 ≤ x ≤ ε n L −13 3n . In fact, by noting varS t n,n ≥ B 2 n 2, for t = 1, 2, whenever 1 ≤ x ≤ ε n L −13 3n and n sufficient large, 3.10 follows immediately from 3.1. The proof of Lemma 3.1 is now complete. ƒ Lemma 3.1 will be used to establish Cra ´ mer type large deviation result for truncated random vari- ables under finite moment conditions. Indeed, as a consequence of Lemma 3.1, we have the follow- ing lemma. Lemma 3.2. If EX 2 = 1 and E|X | α+2α ∞, 0 α ≤ 1, then we have lim n →∞ P ¯ S n ≥ x p n 1 − Φx = 1 3.11 and lim n →∞ P max 1 ≤k≤n ¯ S k ≥ x p n 1 − Φx = 2. 3.12 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 ¯ S k are defined as in 2.2. Proof. Write X n,i = X i I |X i | ≤ p n x α − EX i I |X i | ≤ p n x α and B 2 n = n VarX I |X | ≤ p n x α . It is easy to check that, for R = 2 max{x, 1}B n , n X i=1 E |X n,i | 3 max {e R X n,i , 1 } ≤ C n X i=1 E |X n,i | 3 ≤ C nE |X | 3 , and L 3n := nE |X 1n | 3 B 3 n ∼ n −12 E |X | 3 , uniformly in the range 2 ≤ x ≤ ε n n 1 6 , where C is a constant not depending on x and n. So, by 3.1 in Lemma 3.1, lim n →∞ P ¯ S n − E ¯S n ≥ x p var¯ S n 1 − Φx = 1, 3.13 uniformly in the range 2 ≤ x ≤ ε n n 1 6 . Now, by noting |η n | := r n var¯ S n 1 − p n x EX I |X | ≤ p n x α − 1 ≤ O1 h EX 2 I |X | ≥ p n x α + p n x E |X |I |X | ≥ p n x α i = ox p n, it follows from 3.13 and then 3.9 that lim n →∞ P ¯ S n ≥ x p n 1 − Φx = lim n →∞ P ¯ S n − E ¯S n ≥ x p var¯ S n 1 + η n 1 − Φx = 1, 1188 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. ƒ Lemma 3.3. Suppose that EX 2 = 1 and E|X | 3 ∞. i. For any 0 h 1, λ 0 and θ 0, f h := Ee λhX −θ h 2 X 2 = 1 + λ 2 2 − θ h 2 + O λ,θ h 3 E |X | 3 , 3.14 where |O λ,θ | ≤ e λ 2 θ + 2λ + θ + λ + θ 3 e λ . ii. For any 0 h 1, λ 0 and θ 0, Ee λh max 1 ≤k≤n S k −θ h 2 V 2 n = f n h + n −1 X k=1 f k h ϕ n −k h, 3.15 where ϕ k h = Ee −θ h 2 V 2 k 1 − e λh max 1 ≤ j≤k S j I max 1 ≤ j≤k S j . iii. Write b = x p n. For any λ 0 and θ 0, there exists a sufficient small constant b that may depend on λ and θ such that, for all 0 b b , Ee λb max 1 ≤k≤n S k −θ b 2 V 2 n ≤ C exp λ 2 2 − θ x 2 + O λ,θ x 3 E |X | 3 p n , 3.16 where C is a constant not depending on x and n. Proof. We only prove 3.15 and 3.16. The result 3.14 follows from 2.7 of Shao 1999. Set γ −1 h = 0, γ h = 1, γ n h = Ee λh max{0,max 1 ≤k≤n S k }−θ h 2 V 2 n , for n ≥ 1, Φh, z = ∞ X n=0 γ n h z n , Ψh, z = 1 − f hzΦh, z. Also write ¯ γ n h = γ n h − f hγ n −1 h, b S k = max 1 ≤ j≤k S j and f ∗ h = Ee λh|X |−θ h 2 X 2 . Note that f ∗ h ≤ e λ 2 4θ and γ n h ≤ f ∗ h n by independence of X j . It is readily seen that, for all |z| min {1, 1 f ∗ h}, Ψh, z = 1 + ∞ X n=0 γ n+1 h z n+1 − ∞ X n=0 f h γ n h z n+1 = 1 + ∞ X n=0 γ n+1 h − f hγ n h z n+1 = ∞ X n=0 ¯ γ n h z n . 1189 This, together with f h ≤ f ∗ h, implies that, for all z satisfying |z| min{1, 1| f h|}, [1 − f hz] −1 Ψh, z = ∞ X m=0 ∞ X n=0 f h m ¯ γ n h z m+n = ∞ X n=0 h n X k=0 f k h ¯ γ n −k h i z n . 3.17 By virtue of 3.17 and the identity Φh, z = [1 − f hz] −1 Ψh, z, for all n ≥ 0, γ n h = n X k=0 f k h ¯ γ n −k h. 3.18 Now, by noting that λh max 1 ≤k≤n S k − θ h 2 V 2 n = λhX 1 − θ h 2 X 2 1 + max{0, λhS 2 − X 1 , ..., λhS n − X 1 } − θ h 2 V 2 n − X 2 1 , it follows easily from 3.18 and the i.i.d. properties of X j , j ≥ 1 that Ee λh max 1 ≤k≤n S k −θ h 2 V 2 n = f h γ n −1 h = n X k=1 f k h ¯ γ n −k h. This, together with the facts that ¯ γ h = 1 and ¯ γ k h = γ k h − f hγ k −1 h = Ee λhb S k −θ h 2 V 2 k I b S k ≥0 + Ee −θ h 2 V 2 k I b S k − Ee λhb S k −θ h 2 V 2 k = ϕ k h, for 1 ≤ k ≤ n − 1, gives 3.15. We next prove 3.16. In view of 3.14 and 3.15, it is enough to show that n X k=1 f −k bϕ k b ≤ C, 3.19 where C is a constant not depending on x and n. In order to prove 3.19, let ε be a constant such that 0 ε min{1, λ 2 6θ }, and t 0 sufficiently small such that ǫ 1 := ǫ t − 2t 3 2 1 + E|X | 3 e t 0. First note that ϕ k b = Ee −θ b 2 V 2 k 1 − e λbb S k I b S k ≤ P V 2 k ≤ 1 − ǫ k + e −1−ǫ θ k b 2 E 1 − e λbb S k I b S k . 3.20 1190 It follows from the inequality e x − 1 − x ≤ |x| 3 2 e x ∨0 and the iid properties of X j that P V 2 k ≤ 1 − ǫ k = Pk − V 2 k ≥ ǫ k ≤ e −ǫ t k Ee t 1−X 2 k ≤ e −ǫ t o k 1 + t 3 2 E |1 − X 2 | 3 2 e t k ≤ exp − ǫ t k + 2t 3 2 1 + E|X | 3 ke t = e −ǫ 1 k . As in the proof of Lemma 1 of Aleshkyavichene1979 [see 26 and 28 there], we have E 1 − e λbb S k I b S k = λb 1 p 2 πk + r k + O λ 2 b 2 p k ≤ C b k −12 + λb|r k |, for all k ≥ 1 and sufficient small b, where P ∞ k=1 |r k | = O1 and C is a constant not depending on x and n. Taking these estimates into 3.20, we obtain ϕ k b ≤ e −ǫ 1 k + b C k −12 + λ|r k | e −1−ǫ θ k b 2 , for sufficient small b. On the other hand, it follows easily from 3.14 that there exists a sufficient small b such that for all 0 b ≤ b f b = Ee λbX −θ bX 2 ≥ 1 + λ 2 2 − θ − ǫ θ 2b 2 ≥ e λ 2 2−θ −ǫ θ b 2 , and also f b ≥ e −ǫ 1 2 . Now, by recalling ε λ 2 6θ and using the relation [see 39 in Nagaev 1969] 1 p π ∞ X k=1 z k p k = 1 p 1 − z + ∞ X k=0 ρ k z k = 1 p 1 − z + O1, |z| 1, where ρ k = Ok −32 , simple calculations show that there exists a sufficient small b such that for all 0 b ≤ b n X k=1 f −k bϕ k b ≤ n X k=1 e −ǫ 1 k 2 + b n X k=1 C k −12 + λ|r k | e 2ǫ θ −λ 2 2k b 2 ≤ 1 1 − e −ε 1 2 + C b n X k=1 k −12 e −ǫ θ k b 2 + λb ∞ X k=1 |r k | ≤ C 1 + C p πb p 1 − e −ε θ b 2 ≤ C 2 , 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. ƒ

3.2 Proofs of propositions.