Journal of Inequalities and Applications This Provisional PDF corresponds to the article as it appeared upon acceptance Fully formatted PDF and full text (HTML) versions will be made available soon A note on the almost sure limit theorem for self-normalized partial sums of random variables in the domain of attraction of the normal law Journal of Inequalities and Applications 2012, 2012:17 doi:10.1186/1029-242X-2012-17 Qunying Wu (wqy666@glite.edu.cn) ISSN Article type 1029-242X Research Submission date August 2011 Acceptance date 20 January 2012 Publication date 20 January 2012 Article URL http://www.journalofinequalitiesandapplications.com/content/2012/1/ This peer-reviewed article was published immediately upon acceptance It can be downloaded, printed and distributed freely for any purposes (see copyright notice below) For information about publishing your research in Journal of Inequalities and Applications go to http://www.journalofinequalitiesandapplications.com/authors/instructions/ For information about other SpringerOpen publications go to http://www.springeropen.com © 2012 Wu ; licensee Springer This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited A note on the almost sure limit theorem for self-normalized partial sums of random variables in the domain of attraction of the normal law Qunying Wu1,2 College of Science, Guilin University of Technology, Guilin 541004, P R China Guangxi Key Laboratory of Spatial Information and Geomatics, Guilin 541004, P.R China Email address: wqy666@glite.edu.cn Abstract Let X, X1 , X2 , be a sequence of independent and identically distributed random variables in the domain of attraction of a normal distribution A universal result in almost sure limit theorem for the self-normalized partial sums S n /Vn is established, where S n = n i=1 Xi , Vn = n i=1 Xi2 Mathematical Scientific Classification: 60F15 Keywords: domain of attraction of the normal law; self-normalized partial sums; almost sure central limit theorem Introduction Throughout this article, we assume {X, Xn }n∈N is a sequence of independent and identically distributed (i.i.d.) random variables with a non-degenerate distribution function F For each n ≥ 1, the symbol S n /Vn denotes selfnormalized partial sums, where S n = n i=1 Xi , Vn = n i=1 Xi2 We say that the random variable X belongs to the domain of attraction of the normal law, if there exist constants an > 0, bn ∈ R such that S n − bn d −→ N, an where N is the standard normal random variable We say that {Xn }n∈N satisfies the central limit theorem (CLT) (1) It is known that (1) holds if and only if x2 P(|X| > x) = x→∞ EX I(|X| ≤ x) lim (2) In contrast to the well-known classical central limit theorem, Gine et al [1] obtained the following self-normalized d version of the central limit theorem: (S n − ES n )/Vn −→ N as n → ∞ if and only if (2) holds Brosamler [2] and Schatte [3] obtained the following almost sure central limit theorem (ASCLT): Let {Xn }n∈N be i.i.d random variables with mean 0, variance σ2 > and partial sums S n Then lim n→∞ with dk = 1/k and Dn = n k=1 Dn n dk I k=1 Sk √ < x = Φ(x) a.s for all x ∈ R, σ k (3) dk , where 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 Cs ki [7], Hă rmann [8], Wu [9, 10], and Ye and Wu [11] Huang and Zhang [12] and Zhang a o and Yang [13] obtained ASCLT results for self-normalized version 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, p 35]) shows that the large the weight sequence {dk ; k ≥ 1} in (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 of Schatte [3], Equation (3) fails for weight dk = The optimal weight sequence remains unknown The purpose of this article is to study and establish the ASCLT for self-normalized 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 dk = k−1 exp(ln k)α ), ≤ α < 1/2 Our theorem is formulated in a more general setting Theorem 1.1 Let {X, Xn }n∈N be a sequence of i.i.d random variables in the domain of attraction of the normal law with mean zero Suppose ≤ α < 1/2 and set dk = exp(lnα k) , Dn = k n dk (4) k=1 Then lim n→∞ Dn n dk I k=1 Sk ≤ x = Φ(x) a.s for any x ∈ R Vk (5) 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 {dk }k∈N by any {dk }k∈N such that ≤ ∞ ∗ ∗ dk ≤ dk , k=1 dk = ∞ Remark 1.3 Our results not only give substantial improvements for weight sequence in theorem 1.1 obtained by Huang [12] but also removed the condition nP(|X1 | > ηn ) ≤ c(log n)ε0 , < ε0 < in theorem 1.1 of [12] Remark 1.4 If EX < ∞, then X is in the domain of attraction of the normal law Therefore, the class of random variables in Theorems 1.1 is of very broad range Remark 1.5 Essentially, the open problem should be whether Theorem 1.1 holds for 1/2 ≤ α < remains open Proofs In the following, an ∼ bn denotes limn→∞ an /bn = The symbol c stands for a generic positive constant which may differ from one place to another Furthermore, the following three lemmas will be useful in the proof, and the first is due to [15] Lemma 2.1 Let X be a random variable with EX = 0, and denote l(x) = EX I{|X| ≤ x} The following statements are equivalent: (i) X is in the domain of attraction of the normal law (ii) x2 P(|X| > x) = o(l(x)) (iii) xE(|X|I(|X| > x)) = o(l(x)) (iv) E(|X|α I(|X| ≤ x)) = o(xα−2 l(x)) for α > Lemma 2.2 such that Let {ξ, ξn }n∈N be a sequence of uniformly bounded random variables If exist constants c > and δ > |Eξk ξ j | ≤ c then lim n→∞ Dn k j δ , for ≤ k < j, (6) n dk ξk = a.s., k=1 where dk and Dn are defined by (4) (7) Proof Since     E   n 2    dk ξk  ≤   k=1 n 2 dk Eξk + k=1 dk d j |Eξk ξ j | 1≤k< j≤n n 2 dk Eξk + = k=1 dk d j |Eξk ξl | + 1≤k< j≤n; j/k≥ln2/δ Dn dk d j |Eξk ξl | 1≤k< j≤n; j/k such that |ξk | ≤ c for any k Noting that exp(lnα x) = exp( x α(ln u)α−1 du), u we have exp(lnα x), α < is a slowly varying function at infinity Hence, n ∞ exp(2 lnα k) exp(2 lnα k) ≤c < ∞ k k2 k=1 T n1 ≤ c k=1 By (6), T n2 ≤ c dk d j 1≤k< j≤n; j/k≥ln2/δ Dn k j δ dk d j ≤c 1≤k< j≤n; j/k≥ln2/δ Dn ln2 Dn ≤ cD2 n ln2 Dn (9) On the other hand, if α = 0, we have dk = e/k, Dn ∼ e ln n, hence, for sufficiently large n, n T n3 ≤ c k=1 k k ln2/δ Dn j=k D2 ≤ cDn ln ln Dn ≤ n j ln Dn (10) If α > 0, note that n ln n exp(lnα x) dx = exp(yα )dy x ln n − α −α ∼ exp(yα ) + y exp(yα ) dy α Dn ∼ ln n = 1−α y exp(yα ) dy α = ln1−α n exp(lnα n), n → ∞ α This implies ln Dn ∼ lnα n, exp(lnα n) ∼ αDn (ln Dn ) 1−α α , ln ln Dn ∼ α ln ln n (11) Thus combining |ξk | ≤ c for any k, n T n3 ≤ c dk k=1 dj 1≤k< j≤n; j/k and ε1 := 1/(2α) − > Thus, for sufficiently large n, we get T n3 ≤ c Let T n := Dn n k=1 D2 D2 D2 ln ln Dn n n n ≤ = (ln Dn )1/(2α) (ln Dn )(1−2α)/(2α) (ln Dn )1/(2α) (ln Dn )1+ε1 (12) dk ξk , ε2 := min(1, ε1 ) Combining (8)-(12), for sufficiently large n, we get ET n ≤ c (ln Dn )1+ε2 By (11), we have Dn+1 ∼ Dn Let < η < ε2 /(1 + ε2 ), nk = inf{n; Dn ≥ exp(k1−η )}, then Dnk ≥ exp(k1−η ), Dnk −1 < exp(k1−η ) Therefore 1≤ Dnk Dnk −1 ∼ < → 1, 1−η ) exp(k exp(k1−η ) that is, Dnk ∼ exp(k1−η ) Since (1 − η)(1 + ε2 ) > from the definition of η, thus for any ε > 0, we have ∞ ∞ k=1 ∞ ET nk ≤ c P(|T nk | > ε) ≤ c k=1 k=1 < ∞ k(1−η)(1+ε2 ) By the Borel-Cantelli lemma, T nk → a.s Now for nk < n ≤ nk+1 , by |ξk | ≤ c for any k, |T n | ≤ |T nk | + c Dnk nk+1 di ≤ |T nk | + c i=nk +1 Dnk+1 − → a.s Dnk from Dnk+1 Dnk ∼ exp((k+1)1−η ) exp(k1−η ) = exp(k1−η ((1 + 1/k)1−η − 1)) ∼ exp((1 − η)k−η ) → I.e., (7) holds This completes the proof of Lemma 2.2 Let l(x) = EX I{|X| ≤ x}, b = inf{x ≥ 1; l(x) > 0} and η j = inf s; s ≥ b + 1, l(s) ≤ j s2 for j ≥ By the definition of η j , we have jl(η j ) ≤ η2 and jl(η j − ε) > (η j − ε)2 for any ε > It implies that j nl(ηn ) ∼ η2 , as n → ∞ n (13) For every ≤ i ≤ n, let n n i=1 i=1 Lemma 2.3 ¯2 Xni ¯ ¯2 Xni , Vn = ¯ ¯ Xni = Xi I(|Xi | ≤ ηn ), S n = Suppose that the assumptions of Theorem 1.1 hold Then   n   S − ES ¯k   ¯k   lim ≤ x = Φ(x) a.s for any x ∈ R, dk I      kl(ηk ) n→∞ Dn k=1   k    k n                I  (|Xi | > ηk ) − EI  (|Xi | > ηk ) = a.s.,      lim dk           n→∞ Dn i=1 i=1 k=1 lim n→∞ Dn n k=1     dk  f   ¯2   Vk      − Ef     kl(ηk )  ¯   Vk      = a.s.,     kl(ηk ) (14) (15) (16) where dk and Dn are defined by (4) and f is a non-negative, bounded Lipschitz function Proof ¯ By the cental limit theorem for i.i.d random variables and VarS n ∼ nl(ηn ) as n → ∞ from EX = 0, Lemma 2.1 (iii), and (13), it follows that ¯ ¯ S n − ES n nl(ηn ) d −→ N, as n → ∞, where N denotes the standard normal random variable This implies that for any g(x) which is a non-negative, bounded Lipschitz function    S n − ES n  ¯  ¯   −→ Eg(N), as n → ∞,   Eg     nl(ηn ) Hence, we obtain    S k − ES k  ¯  ¯   = Eg(N)   dk Eg     kl(ηk ) k=1 n lim n→∞ Dn from the Toeplitz lemma On the other hand, note that (14) is equivalent to lim n→∞ Dn    S k − ES k  ¯  ¯   = Eg(N) a.s   dk g     kl(ηk ) k=1 n from Theorem 7.1 of [16] and Section of [17] Hence, to prove (14), it suffices to prove lim n→∞ Dn        S k − ES k  ¯  ¯  ¯  ¯   − Eg  S k − ES k  = a.s.,       dk g            kl(ηk ) kl(ηk ) k=1 n (17) for any g(x) which is a non-negative, bounded Lipschitz function For any k ≥ 1, let      S k − ES k  ¯  ¯  ¯ ¯    − Eg  S k − ES k       ξk = g         kl(ηk ) kl(ηk )    S j −ES j − k (Xi −EXi )I(|Xi |≤η j )    ¯ ¯ i=1   ¯ ¯ S k −ES k  are independent and g(x) is a non-negative,    √ For any ≤ k < j, note that g √ and g       jl(η j ) kl(ηk ) bounded Lipschitz function By the definition of η j , we get,      ¯    S k − ES k   S j − ES j  ¯  ¯  ¯ g  ,g       |Eξk ξ j | = Cov           jl(η j ) kl(ηk )    k   ¯   S j − ES j − (Xi − EXi )I(|Xi | ≤ η j )      ¯     ¯      ¯  ¯  ¯   S k − ES k   S j − ES j   i=1     ,g  − g g         = Cov                      jl(η j ) jl(η j ) kl(ηk )       E (Xi − EXi )I(|Xi | ≤ η j ) i=1 ≤c =c k jl(η j ) k j kEX I(|X| ≤ η j ) ≤c jl(η j ) 1/2 By Lemma 2.2, (17) holds Now we prove (15) Let      Zk = I    k i=1         − EI    (|Xi | > ηk )     k i=1      (|Xi | > ηk )   for any k ≥ It is known that I(A ∪ B) − I(B) ≤ I(A) for any sets A and B, then for ≤ k < j, by Lemma 2.1 (ii) and (13), we get P(|X| > η j ) = o(1) l(η j ) η2 j = o(1) j Hence    j   k                I  (|Xi | > ηk ) , I  (|Xi | > η j )      |EZk Z j | = Cov           i=1 i=1   j    j   k                             (|Xi | > η j ) = Cov I  (|Xi | > ηk ) , I  (|Xi | > η j ) − I                i=1 i=1 i=k+1   j   j                 (|Xi | > η j ) ≤ E I  (|Xi | > η j ) − I          i=1 i=k+1   k        (|Xi | > η j ) ≤ kP(|X| > η j )   ≤ EI     i=1 k ≤ j By Lemma 2.2, (15) holds Finally, we prove (16) Let  ¯2   V      ζk = f  k  − E f   kl(ηk )  ¯2   Vk          kl(ηk ) For ≤ k < j,     |Eζk ζ j | = Cov  f    ¯2   Vk     ,     kl(ηk )  ¯   V j      f    jl(η )   j for any k ≥ (18)          = Cov  f       k E i=1 ≤c  ¯2   Vk     ,     kl(ηk )  ¯2   Vj      f     jl(η )  − j Xi2 I(|Xi | ≤ η j ) =c jl(η j )    ¯2 k  V − X I(|Xi | ≤ η j )    j    i     i=1      f       jl(η j )       kEX I(|X| ≤ η j ) kl(η j ) =c jl(η j ) jl(η j ) k =c j By Lemma 2.2, (16) holds This completes the proof of Lemma 2.3 Proof of Theorem 1.1 For any given < ε < 1, note that   Sk   ≤ x ≤ I I   Vk Sk I ≤x ≤ Vk     I   ¯ Sk (1 + ε)kl(ηk ) ¯ Sk (1 − ε)kl(ηk )        ¯2   ≤ x + I Vk > (1 + ε)kl(ηk ) + I             + I V < (1 − ε)kl(η ) + I   ¯k  ≤ x   k   k i=1 k i=1      |Xi | > ηk ) , for x ≥ 0,        |Xi | > ηk ) , for x < 0,   and   Sk   I ≤ x ≥ I   Vk   Sk   I ≤ x ≥ I   Vk ¯ Sk (1 − ε)kl(ηk ) ¯ Sk (1 + ε)kl(ηk )        − I V < (1 − ε)kl(η ) − I   ¯k  ≤ x   k          − I V > (1 + ε)kl(η ) − I   ¯k   ≤ x  k   k i=1 k i=1      |Xi | > ηk ) , for x ≥ 0,        |Xi | > ηk ) , for x <   Hence, to prove (5), it suffices to prove lim n→∞ Dn n     dk I    ¯ Sk k=1 lim n→∞ Dn lim n→∞ Dn kl(ηk )  n     dk I    k=1  √  √   ≤ ± εx = Φ( ± εx) a.s.,   k i=1      |Xi | > ηk ) = a.s.,   (19) (20) n ¯2 dk I(Vk > (1 + ε)kl(ηk )) = a.s., k=1 (21) lim n→∞ Dn n ¯2 dk I(Vk < (1 − ε)kl(ηk )) = a.s (22) k=1 by the arbitrariness of ε > Firstly, we prove (19) Let < β < 1/2 and h(·) be a real function, such that for any given x ∈ R, I(y ≤ √ ± εx − β) ≤ h(y) ≤ I(y ≤ √ ± εx + β) By EX = 0, Lemma 2.1 (iii) and (13), we have ¯ |ES k | = |kEXI(|X| ≤ ηk )| = |kEXI(|X| > ηk )| ≤ kE|X|I(|X| > ηk ) = o( kl(ηk )) This, combining with (14), (23) and the arbitrariness of β in (23), (19) holds By (15), (18) and the Toeplitz lemma, 0≤ Dn ≤ Dn n k=1 n      dk I         |Xi | > ηk ) ∼   Dn i=1 k n k=1      dk EI    k i=1      |Xi | > ηk )   dk kP(|X| > ηk ) → a.s k=1 That is (20) holds Now we prove (21) For any µ > 0, let f be a non-negative, bounded Lipschitz function such that I(x > + µ) ≤ f (x) ≤ I(x > + µ/2) ¯ ¯2 Form EVk = kl(ηk ), Xni is i.i.d., Lemma 2.1 (iv), and (13), µ ¯2 ¯2 µ ¯2 P Vk > + kl(ηk ) = P Vk − EVk > kl(ηk ) 2 ¯2 ¯2 E(Vk − EVk )2 EX I(|X| ≤ ηk ) ≤c ≤c k2 l2 (ηk ) kl2 (ηk ) o(1)ηk = o(1) → = kl(ηk ) 10 (23) Therefore, from (16) and the Toeplitz lemma,  ¯2  n  V    ¯2   dk I Vk > (1 + µ)kl(ηk ) ≤ dk f  k    Dn k=1 kl(ηk ) k=1  ¯2  n n  V  1   ¯2 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