A NOTE FOR EXTENSION OF ALMOST SURE CENTRAL LIMIT THEORY 7 0 YU MIAO AND GUANGYU YANG 0 2 n Abstract. Ho¨rmann(2006)gaveanextensionofalmostsurecentrallimittheorem a for bounded Lipschitz 1 function. In this paper, we show that his result of almost J sure central limit theorem is also hold for any Lipschitz function under stronger 6 conditions. ] R P . h t 1. Introduction a m The classical results on the almost sure central limit theorem (ASCLT) dealt with [ partial sums of random variables. A general pattern is that, if X1,X2,... be a 1 sequence of independent random variables with partial sums S = X + + X n 1 n v ··· 3 satisfying (S b )/a L H for some sequences a > 0, b R and some distribution n n n n n 8 function H, th−en under−→some mild conditions we have ∈ 1 1 n 0 1 1 lim I (S b )/a x = H(x) a.s. 7 k k k 0 n→∞ logn X k n − ≤ o k=1 / h for any continuity point x of H. t a Several papers have dealt with logarithmic limit theorems of this kind and the m above relation has been extended in various directions. Fahrner and Stadtmu¨ller : v [5] gave an almost sure version of a maximum limit theorem. Berkes and Horváth i X [2] obtained a strong approximation for the logarithmic average of sample extremes. r Berkes and Csáki [1] showed that not only the central limit theorem, but every weak a limittheoremforindependent randomvariableshasananalogousalmost sureversion. For stationary Gaussian sequences with covariance r , Csáki and Gonchigdanzan [3] n proved an almost sure limit theorem for the maxima of the sequences under the condition r logn(loglogn)1+ε = O(1). For some dependent random variables, Peligrad n and Shao [7] and Dudzin´ski [4] obtained corresponding results about the almost sure central limit theorem. Date: May 26, 2006. 2000 Mathematics Subject Classification. 60F05. Key words and phrases. Almost sure central limit theorem; Lipschitz function; logarithmic limit theorems. 1 2 YU MIAO ANDGUANGYUYANG Recently, Hörmann [6] gave an extension of almost sure central limit theory under some regularity condition as the following form: N S ∞ lim D−1 d f k b = f(x)dH(x) a.s. (1.1) N→∞ N X k (cid:16)ak − k(cid:17) Z−∞ k=1 where f is a bounded Lipschitz 1 function and D = N d , d is a sequence N k=1 k { k}k≥1 of positive constants. Using his method, we will showPthat for any Lipschitz function f, (1.1) holds under some additional conditions. At first, we give our main result. Theorem 1.1. Let X ,X ,... be independent random variables with partial sums S 1 2 n and assume that S : (C ) For some numerical sequences a > 0 and b , we have n b L H, 1 n n n a − −→ n where H is some (possibly degenerate) distribution function. : (C ) kd = O(1) and for some 0 < α < 1, d kα is eventually non-increasing. 2 k k D : (C ) For some ρ > 0, d = O k . 3 k (cid:16)k(logDk)ρ(cid:17) : (C ) There exist positive constants C, β, such that a /a C(k/l)β (1 k 4 k l ≤ ≤ ≤ l). Furthermore, for some 0 < r < ρ, S µ E n b = O(1), for some integer µ (2 4/(ρ r)). (1.2) n (cid:12)a − (cid:12) ≥ ∨ − n (cid:12) (cid:12) (cid:12) (cid:12) Then for any Lipschitz function f on the real line, we have (1.1). Remarks 1.2. Obviously, for any bounded Lipschitz 1 function f, under the above assumptions, the equation (1.1) holds, i.e. we can obtain Theorem 1 in [6]. 2. Proof of Theorem 1.1 In this section, we will give the proof of Theorem 1.1, according to the process of Hörmann in [6]. Lemma 2.1. (See Lemma 1 in [6] ) Let (D ) be a summation procedure, then the N condition (C ) of Theorem 1.1 implies that D = o(Nε) for any ε > 0. 3 N Lemma 2.2. Assume that condition (C ) of Theorem 1.1 is satisfied and b = 0. 4 n Then for every Lipschitz function f : R R there exists constant c > 0 such that → S S Cov f k ,f l c(k/l)β (1 k l), (2.1) (cid:12) (cid:16) (cid:16)ak(cid:17) (cid:16)al(cid:17)(cid:17)(cid:12) ≤ ≤ ≤ (cid:12) (cid:12) (cid:12) (cid:12) where β is the same as in (C ). 4 A NOTE FOR EXTENSION OF ALMOST SURE CENTRAL LIMIT THEORY 3 Proof. Firstly, we assume f(0) = 0. Denoting f the Lipschitz constant of f, we k k get, by using the independence of S and S S , k l k − S S S S S S k l k l l k Cov f ,f = Cov f ,f f − (cid:12) (cid:16) (cid:16)ak(cid:17) (cid:16)al(cid:17)(cid:17)(cid:12) (cid:12) (cid:16) (cid:16)ak(cid:17) (cid:16)al(cid:17)− (cid:16) al (cid:17)(cid:17)(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) S S (cid:12)S (cid:12) S S S S(cid:12) S E f k f l f l − k +E f k E f l f l − k ≤ (cid:12) (cid:16)ak(cid:17)h (cid:16)al(cid:17)− (cid:16) al (cid:17)i(cid:12) (cid:12) (cid:16)ak(cid:17)(cid:12) (cid:12) (cid:16)al(cid:17)− (cid:16) al (cid:17)(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) a S2 a S (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) f 2 kE k + f 2 k(E k )2 ≤k k al ha2ki k k al haki a S2 S2 2 f 2 kE k 2C f 2E k (k/l)β, ≤ k k al ha2ki ≤ k k ha2ki where the last inequality is due to condition (C ). Since the equation (1.2), we can 4 S2 take c > 0 such that for any k 1, 2C f 2E k c. So (2.1) is obtained. ≥ k k ha2i ≤ k If f(0) = 0, we can define a function g, such that g(x) = f(x) f(0), then g is a 6 − Lipschitz function and g(0) = 0. And noting that, S S S S k l k l Cov f ,f = Cov g ,g (cid:16) (cid:16)ak(cid:17) (cid:16)al(cid:17)(cid:17) (cid:16) (cid:16)ak(cid:17) (cid:16)al(cid:17)(cid:17) we complete the proof of the lemma. (cid:3) Remarks 2.3. It is obvious to see that if we replace β by any 0 < β′ < β, the Lemma 2.2 also holds. Hence, without loss of generality, we can assume that β is the same as α in condition (C ) of Theorem 1.1. 2 Next we will use the following notations, S S S S S S ξ := f l Ef l , ξ := f l − k Ef l − k . (2.2) l k,l (cid:16)al(cid:17)− (cid:16)al(cid:17) (cid:16) al (cid:17)− (cid:16) al (cid:17) Lemma 2.4. Assume that condition (C ) of Theorem 1.1 is satisfied and b = 0, 4 n and define ξ and ξ as in (2.2). If d ,k 1 are arbitrary positive weights, then l k,l k we have for any k m n and p N{, p ≥µ, } ≤ ≤ ∈ ≤ n n p p/2 E d (ξ ξ ) E d2l , (cid:12)X l l − k,l (cid:12) ≤ p(cid:16)X l (cid:17) (cid:12)l=m (cid:12) l=m (cid:12) (cid:12) 2κ 1 p/2 where E = c Cp f p 1+ and κ = 2β. p p k k h(cid:16) κ (cid:17)∨(cid:16) κ(cid:17)i 4 YU MIAO ANDGUANGYUYANG Proof. Let Q(l) = Q(k,l) = ξ ξ , then l k,l − S S S S S S p E Q(l) p =E f l f l − k E f l f l − k | | (cid:12) (cid:16)al(cid:17)− (cid:16) al (cid:17)− h (cid:16)al(cid:17)− (cid:16) al (cid:17)i(cid:12) (cid:12) (cid:12) (cid:12) S S p (cid:12) f p(a /a )pE | k| +E | k| k l ≤k k (cid:16) ak (cid:16) ak (cid:17)(cid:17) S S p Cp f pE | k| +E | k| (k/l)pβ ≤ k k (cid:16) ak (cid:16) ak (cid:17)(cid:17) c Cp f p(k/l)pβ, p ≤ k k where C is the same as in condition (C ) and c is a positive constant such that for 4 p S S p all k, E | k| +E | k| c . Thus by the Hölder inequality, we get p (cid:16) ak (cid:16) ak (cid:17)(cid:17) ≤ n n n p E d (ξ ξ ) d d (E Q(l ) p E Q(l ) p)1/p (cid:12)X l l − k,l (cid:12) ≤ X ··· X l1··· lp | 1 | ··· | p | (cid:12)l=m (cid:12) l1=m lp=m (cid:12) (cid:12) n n c Cp f pkpβ d d l−β l−β ≤ p k k ··· l1··· lp 1 ··· p X X l1=m lp=m n p =c Cp f pkpβ d l−β p l k k (cid:16)X (cid:17) l=m n n p/2 p/2 c Cp f pmpβ d2l l−2β−1 . ≤ p k k (cid:16)X l (cid:17) (cid:16)X (cid:17) l=m l=m For m 2, it is easy to see that ≥ n ∞ p/2 p/2 mpβ l−2β−1 mpβ l−2β−1dl (cid:16)X (cid:17) ≤ (cid:16)Zm−1 (cid:17) l=m m pβ 1 p/2 2κ p/2 , ≤(cid:16)m 1(cid:17) (cid:16)2β(cid:17) ≤ (cid:16) κ (cid:17) − where κ := 2β. Similarly, we get for m = 1 n p/2 1 p/2 l−2β−1 1+ . (cid:16)X (cid:17) ≤ (cid:16) κ(cid:17) l=1 Hence, we have n n p p/2 E d (ξ ξ ) c Cp f pτ(κ)p/2 d2l , (cid:12)X l l − k,l (cid:12) ≤ p k k (cid:16)X l (cid:17) (cid:12)l=m (cid:12) l=m (cid:12) (cid:12) where τ(κ) := 2κ 1+ 1 . This completes the proof of our result. (cid:3) h(cid:16) κ (cid:17)∨(cid:16) κ(cid:17)i A NOTE FOR EXTENSION OF ALMOST SURE CENTRAL LIMIT THEORY 5 Lemma 2.5. Assume that conditions (C ) (C ) of Theorem 1.1 are satisfied. Fur- 2 4 − ther let b = 0 in condition (C ) and f be a Lipschitz function. Then for every p µ n 4 and p N we have ≤ ∈ N S S p k β p/2 E d f k Ef k C d d , (2.3) k p k l (cid:12)X (cid:16) (cid:16)ak(cid:17)− (cid:16)ak(cid:17)(cid:17)(cid:12) ≤ (cid:16) X (cid:16)l(cid:17) (cid:17) (cid:12)k=1 (cid:12) 1≤k≤l≤N (cid:12) (cid:12) where C > 0 is a constant and β is the same as in (C ). p 4 Proof. At first, we set C = (4γ)p2 and p n l V := d l−β d kβ , (1 m n). m,n l k X (cid:16)X (cid:17) ≤ ≤ l=m k=1 For obtaining our result, it is enough to show that the following claim, ”if the number γ is chosen large enough, then n p E d ξ C (V )p/2, for all 1 m n, (2.4) k k p m,n (cid:12)X (cid:12) ≤ ≤ ≤ (cid:12)k=m (cid:12) (cid:12) (cid:12) where ξ is defined as in (2.2).” k We will use induction on p to show (2.4). By Lemma 2.2, we have n 2 E d ξ 2 d d Eξ ξ 2c d d (k/l)β 2cV . k k k l k l k l m,n (cid:12)X (cid:12) ≤ X | | ≤ X ≤ (cid:12)k=m (cid:12) m≤k≤l≤n m≤k≤l≤n (cid:12) (cid:12) Hence if we choose γ so large that (4γ)4 2c, then (2.4) holds for p = 2. ≥ Assume now that (2.4) is true for p 1 2. From kd = O(1) it follows that there k − ≥ is a positive constant A such that l d kβ Alβ. Then we get for V γ as k=1 k ≥ m,n ≤ the proof of Lemma 2.4, there existPs a constant A such that p n n n p E d ξ d d (E ξ p E ξ p)1/p (cid:12)X k k(cid:12) ≤ X ··· X k1··· kp | k1| ··· | kp| (cid:12)k=m (cid:12) k1=m kp=m (cid:12) (cid:12) n n A f p d d ≤ pk k ··· k1··· kp X X k1=m kp=m n p =A f p d p k k k (cid:16)X (cid:17) k=m n k p A f pA−p d k−β d lβ . p k l ≤ k k (cid:16)X (cid:16)X (cid:17)(cid:17) k=m l=1 Now choose γ so large that the C (A 1/p f /A)pγp/2. In the case of V γ, we p p m,n ≤ k k ≤ have shown (2.4) is valid. 6 YU MIAO ANDGUANGYUYANG We now want to show that if for any given X γ and the inequality (2.4) holds ≥ for V X, then it will also hold for V 3X/2 and this will show that (2.4) m,n m,n ≤ ≤ holds for any value of V , i.e. complete the induction step. m,n Assume V 3X/2 and set m,n ≤ w n n S +S := d ξ + d ξ , T := d ξ , (m w n). 1 2 k k k k 2 k w,k ≤ ≤ X X X k=m k=w+1 k=w+1 From the discussion of Lemma 4 in Hörmann, S. [6] (2006), and Lemma 2.1, for a fixed m and n we choose w in such a way that V w+1,n V X, V X and = λ [1/2,1]. m,w w+1,n ≤ ≤ V ∈ m,w From the mean value theorem we get Sj Tj j S T ( S j−1 + T j−1) (j 1). (2.5) | 2 − 2| ≤ | 2 − 2| | 2| | 2| ≥ Since condition (C ) and Remarks 2.3, there exists a constant B > 0 such that for 2 all l 1, ≥ l B d kβ l1+βd . k l ≥ X k=1 This also shows that n ld2 BV , for all 1 m n. l ≤ m,n ≤ ≤ X l=m By Lemma 2.4, we get for all j 1, ≥ E S T j F (V )j/2, 2 2 j w+1,n | − | ≤ where F = Bj/2E and E is the constant in Lemma 2.4. j j j From the induction hypothesis in the case of 1 j p 1 and from the validity ≤ ≤ − of (2.4) for V X in the case of j = p, we have m,n ≤ E S j C (V )j/2, (1 j p) (2.6) 1 j m,w | | ≤ ≤ ≤ and E S j C (V )j/2 C λj/2(V )j/2, (1 j p). (2.7) 2 j w+1,n j m,w | | ≤ ≤ ≤ ≤ The remains of the proof are the same as in Lemma 4 in Hörmann, S. [6] (2006), but for completeness, we still give the proof. By C inequality, we have r E T j 2jC λj/2(V )j/2, (1 j p). (2.8) 2 j m,w | | ≤ ≤ ≤ Furthermore, from Hölder inequality the following result is easy, E S j S T S p−j−1 (E S p)j/p(E S T p)1/p(E S p)(p−j−1)/p 1 2 2 2 1 2 2 2 | | | − || | ≤ | | | − | | | C(p−1)/pF1/pλ(p−j)/2(V )p/2. (2.9) ≤ p p m,w A NOTE FOR EXTENSION OF ALMOST SURE CENTRAL LIMIT THEORY 7 The last inequality remains valid, with an extra factor 2p−j−1 on the right hand side, if S p−j−1 on the left hand side is replaced by T p−j−1. Since S and T are 2 2 1 2 | | | | independent, we get p−1 E S +S p E S p +E S p + Gj(E S j Sp−j Tp−j +E S jE T p−j), | 1 2| ≤ | 1| | 2| p | 1| | 2 − 2 | | 1| | 2| X j=1 where Gj denote the combination, i.e., Gj = p![j!(p j)!]−1. Substituting (2.5) (2.9) p p − − (using also the analogue of (2.9) with T p−j−1) in the above inequality and get 2 | | p−1 E S +S p C (V )p/2 1+λp/2 +C−1/pF1/p 2p−jGj(p j)λ(p−j)/2 | 1 2| ≤ p m,w (cid:16) p p X p − j=1 p−1 +C−1 2p−jGjλ(p−j)/2C C . p p j p−j X (cid:17) j=1 Note that C−1/pF1/p const τ(κ)1/2c1/p(4γ)−p p p ≤ · p and C C /C (4γ)−p, (1 j p 1), j p−j p ≤ ≤ ≤ − thus, by λ 1, we have ≤ p−1 C−1/pF1/p 2p−jGj(p j)λ(p−j)/2 const τ(κ)1/2pc1/pγ−p p p p − ≤ · p X j=1 and p−1 C−1 2p−jGjλ(p−j)/2C C const γ−p. p p j p−j ≤ · X j=1 Since λ 1/2 we have shown that for a large γ the relation E S + S p C (1 + 1 2 p ≥ | | ≤ λ)p/2(V )p/2 = C (V )p/2 is true, i.e., for V 3X/2, (2.4) is valid. (cid:3) m,w p m,n m,n ≤ Lemma 2.6. (See Lemma 5 in [6]) Assume the condition (C ) of Theorem 1.1 is 3 satisfied. Then for any α > 0 and any η < ρ, we have k α D2 d d = O N . X k l(cid:16)l(cid:17) (cid:16)(logDN)η(cid:17) 1≤k≤l≤N Proof of Theorem 1.1. Without loss of generality, from Lemma 2.5 and Lemma 2.6, we have, for any ε > 0, p µ and p N, ≤ ∈ N P d ξ > εD c(p,ε)(logD )−pη/2, for N N . k k N N 0 (cid:16)(cid:12)X (cid:12) (cid:17) ≤ ≥ (cid:12)k=1 (cid:12) (cid:12) (cid:12) 8 YU MIAO ANDGUANGYUYANG Since µ (2 4/(ρ r)) for some 0 < r < ρ, we can choose suitable η < ρ and p ≥ ∨ − such that pη > 4. By (C ), we have D /D 1, thus we can choose (N ) such 3 N+1 N j → that D exp √j . Applying Borel-Cantelli lemma, we get Nj ∼ { } Nj lim D−1 d ξ = 0 a.s.. j→∞ Nj X k k k=1 For N N N , we have j j+1 ≤ ≤ N Nj D−1 d ξ D−1 d ξ +2(D /D 1) a.s.. N | k k| ≤ Nj| k k| Nj+1 Nj − X X k=1 k=1 The convergence of the subsequence implies that whole sequence converges a.s., since D /D 1. This complete the proof of the theorem. (cid:3) Nj+1 Nj → References [1] Berkes, I., Csáki, E. A universal result in almost sure central limit theory. Sto- chastic Process and Applications, 2001, 94: 105-134. [2] Berkes, I., Horváth L. The logarithmic average of sample extremes is asymptot- ically normal. Stochastic Process and Applications, 2001, 91(1): 77-98. [3] Csáki, E., Gonchigdanzan, K. Almost sure limit theorems for the maximum of stationary Gaussian sequences. Statistics and Probability Letters, 2002, 58(2): 195-203. [4] Dudzin´ski, M. A note on the almost sure central limit theorem for some dependent random variables. Statistics and Probability Letters, 2003, 61: 31-40. [5] Fahrner, I., Stadtmu¨ller, U. On almost sure max-limit theorems. Statistics and Probability Letters, 1998, 37(3): 229-236. [6] Hörmann, S. An extension of almost sure central limit theory. Statistics and Probability Letters, 2006, 76: 191-202. [7] Peligrad, M., Shao, Q.M. A note on the almost sure central limit theorem for weakly dependent random variables. Statistics and Probability Letters, 1995, 22: 131-136. DepartmentofMathematicsandStatistics,WuhanUniversity,430072Hubei,China and College of Mathematics and Information Science, Henan Normal University, 453007 Henan, China. E-mail address: yumiao728@yahoo.com.cn DepartmentofMathematicsandStatistics,WuhanUniversity,430072Hubei,China. E-mail address: study yang@yahoo.com.cn

A note for extension of almost sure central limit theory

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