Wu and Chen Journal of Inequalities and Applications 23, 23:29 http://www.ournalofinequalitiesandapplications.com/content/23//29 R E S E A R C H Open Access An improved result in almost sure central it theorem for self-normalized products of partial sums Qunying Wu * and Pingyan Chen 2 * Correspondence: wqy666@glut.edu.cn College of Science, Guilin University of Technology, Guilin, 544, P.R. China Full list of author information is available at the end of the article Abstract Let X, X, X 2,...be a sequence of independent and identically distributed random variables in the domain of attraction of the normal law. A universal result in an almost sure it theorem for the self-normalized products of partial sums is established. MSC: 6F5 Keywords: domain of attraction of the normal law; self-normalized partial sums; almost sure central it theorem Introduction Let {X, X n } n N be a sequence of independent and identically distributed i.i.d. positive random variables with a non-degenerate distribution function and EX >.Foreach n, the symbol S n /V n denotes self-normalized partial sums, where S n n X i, Vn 2 n X i 2. We say that the random variable X belongs to the domain of attraction of the normal law if there exist constants a n >,b n R such that S n b n a n d N. Here and in the sequel, N is a standard normal random variable, and d denotes the convergence in distribution. We say that {X n } n N satisfies the central it theorem CLT. It is nown that holds if and only if x x 2 P X > x. 2 EX 2 I X x In contrast to the well-nown classical central it theorem, Gine et al. [7]obtainedthe d following self-normalized version of the central it theorem: S n ES n /V n N as n if and only if 2holds. The it theorem of products n S was initiated by Arnold and Villaseñor []. Their result was generalized by Wu [2], Ye and Wu [22], and Rempala and Wesolowsi [6]who proved that if {X n ; n } is a sequence of i.i.d. positive and finite second moment random variables with EX, Var X σ 2 > and the coefficient of variation γ σ /,then n S i n! n /γ n d e 2N as n. 3 23 Wu and Chen; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/2., which permits unrestricted use, distribution, and reproduction in any medium, provided the original wor is properly cited.
Wu and Chen Journal of Inequalities and Applications 23, 23:29 Page 2 of 2 http://www.ournalofinequalitiesandapplications.com/content/23//29 Recently Pang et al. [4] obtained the following self-normalized products of sums for i.i.d. sequences: Let {X, X n } n N be a sequence of i.i.d. positive random variables with EX >, and assume that X is in the domain of attraction of the normal law. Then n S i n! n /Vn d e 2N as n. 4 Brosamler [4] andschatte[7] obtained the following almost sure central it theorem ASCLT: Let {X n } n N be i.i.d. random variables with mean, variance σ 2 >,andpartial sums S n.then { } S d I σ < x x a.s. for all x R, 5 with d /and n d ;hereandinthesequel,i denotes an indicator function, and x is the standard normal distribution function. Some ASCLT results for partial sums were obtained by Lacey and Philipp [2], Ibragimov and Lifshits [], Miao [3], Beres and Csái [2], Hörmann [9], Wu [8, 9]. Gonchigdanzan and Rempala [8] gaveasclt for products of partial sums. Huang and Pang [], Wu [2], and Zhang and Yang [23] 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., Chandraseharan and Minashisundaram [5], p.35 shows that the larger the weight sequence {d ; } in 5 is, the stronger the relation becomes. By this argument, one should also expect to get stronger results if we use larger weights. It would be of considerable interest to determine the optimal weights. On the other hand, by Theorem of Schatte [7], 5failsforweightd.Theoptimal weight sequence remains unnown. ThepurposeofthispaperistostudyandestablishtheASCLTforself-normalized products of partial sums of random variables in the domain of attraction of the normal law. We show that the ASCLT holds under a fairly general growth condition on d expln α, α < /2. Inthefollowing,weassumethat{X, X n } n N is a sequence of i.i.d. positive random variables in the domain of attraction of the normal law with EX >.Letb,n n /, S X i, V 2 X i 2, S, b i,x i for n. a n b n denotes a n /b n.thesymbolc stands for a generic positive constant which may differ from one place to another. Our theorem is formulated in a general setting. Theorem. Let {X, X n } n N be a sequence of i.i.d. positive random variables in the domain of attraction of the normal law with mean >.Suppose α < /2 and set d explnα, d. 6
Wu and Chen Journal of Inequalities and Applications 23, 23:29 Page 3 of 2 http://www.ournalofinequalitiesandapplications.com/content/23//29 Then d I S /V i x Fx a.s. x R. 7! Here and in the sequel, F is the distribution function of the random variable e 2N. By the terminology of summation procedures, we have the following corollary. Corollary.2 Theorem. remains valid if we replace the weight sequence {d } N by {d } N such that d d, d. Remar.3 Our results give substantial improvements for weight sequence in Theorem. obtained by Zhang and Yang [23]. Remar.4 If X is in the domain of attraction of the normal law, then E X p < for <p <2.Onthecontrary,ifEX 2 <,thenx is in the domain of attraction of the normal law. Therefore, the class of random variables in Theorem. is of very broad range. Remar.5 Essentially, the problem whether Theorem. holds for /2 α <remains open. 2 Proofs Furthermore, the following three lemmas will be useful in the proof, and the first is due to Csörgo et al. [6]. Lemma 2. Let X be a random variable with EX, and denote lxex 2 I{ X x}. The following statements are equivalent. i X is in the domain of attraction of the normal law. ii x 2 P X > xolx. iii xe X I X > x olx. iv E X α I X x ox α 2 lx for α >2. v lx is a slowly varying function at. Lemma 2.2 Let {ξ, ξ n } n N be a sequence of uniformly bounded random variables. If there exist constants c >and δ >such that δ Eξ ξ c for <, 8 then d ξ a.s., 9 where d and are defined by 6.
Wu and Chen Journal of Inequalities and Applications 23, 23:29 Page 4 of 2 http://www.ournalofinequalitiesandapplications.com/content/23//29 Proof Since 2 E d ξ d 2 Eξ 2 +2 < n d d Eξ ξ d 2 Eξ 2 +2 d d Eξ ξ < n;/ ln 2/δ +2 d d Eξ ξ < n;/<ln 2/δ : T n +2T n2 + T n3. By the assumption of Lemma 2.2, there exists a constant c >suchthat ξ c for any.notingthatexpln α xexp x αln u α du, we have that expln α x, α <,isaslowly u varying function at infinity. Hence, T n By 8, T n2 exp2 ln α 2 < n;/ ln 2/δ d d < n;/ ln 2/δ exp2 ln α 2 <. δ d d ln 2 cd2 n ln 2. On the other hand, if α,wehaved e/, e ln n, and hence, for sufficiently large n, T n3 ln 2/δ ln ln D2 n ln 2. 2 D n If < α < /2, then by y α, y, for arbitrary small ε >,thereexistsn such that for y ln n, αy α /α < ε. Therefore expy α + α α y α expy α dy expy α dy This implies expy α + α α y α dy ++ε ln n expy α dy +ε. expy α dy exp y α dy from the arbitrariness of ε. exp y α + α α y α exp y α dy
Wu and Chen Journal of Inequalities and Applications 23, 23:29 Page 5 of 2 http://www.ournalofinequalitiesandapplications.com/content/23//29 Hence, n expln α x dx exp y α dy x exp y α + α α y α exp y α dy y α exp y α dy α α ln α n exp ln α n, n. 3 This implies ln ln α n, exp ln α n α, ln ln D ln α n α ln ln n. α Thus combining ξ c for any, T n3 < n;/<ln 2/δ d d d < ln 2/δ exp exp ln α n ln ln D2 n ln ln ln α/α. ln α n d Since α < /2 implies 2α/2α>andε : /2α >, thus for sufficiently large n, we get D 2 n T n3 ln /2α ln ln ln 2α/2α D 2 n ln /2α D 2 n ln +ε. 4 Let T n : n d ξ, ε 2 : min, ε. Combining -2and4, for sufficiently large n,weget ET 2 n c ln +ε 2. By 3, we have +.Let<η < ε 2 +ε 2, n inf{n; exp η }, then exp η, < exp η. Therefore that is, exp η exp η <, exp η.
Wu and Chen Journal of Inequalities and Applications 23, 23:29 Page 6 of 2 http://www.ournalofinequalitiesandapplications.com/content/23//29 Since η + ε 2 > from the definition of η, thus for any ε >,wehave P T n > ε ETn 2 <. η+ε 2 By the Borel-Cantelli lemma, T n a.s. Now, for n < n n +,by ξ c for any, T n T n + c n + in + Dn+ d i T n + c a.s. from + exp+ η exp η + / η exp η η, i.e., 9 holds. exp η This completes the proof of Lemma 2.2. Let lxex 2 I{ X x}, b inf{x ; lx>} and { η inf s; s b +, ls s } 2 for. By the definition of η,wehavelη η 2 and lη ε>η ε 2 for any ε >.Itimplies that nlη n η 2 n as n. 5 For every i n,let X i X i I X i η, V 2 X i 2, S, b i, X i. Lemma 2.3 Suppose that the assumptions of Theorem. hold. Then { } S, E S, d I x x a.s. for any x R, 6 2lη d I Xi > η EI Xi > η a.s., 7 V d f 2 V 2 Ef a.s., 8 lη lη where d and are defined by 6 and f is a non-negative, bounded Lipschitz function. Proof By the central it theorem for i.i.d. random variables and Var S n,n 2nlη n as n from n b2,n 2n, it follows that S n,n E S n,n 2nlηn d N as n,
Wu and Chen Journal of Inequalities and Applications 23, 23:29 Page 7 of 2 http://www.ournalofinequalitiesandapplications.com/content/23//29 where N denotes the standard normal random variable. This implies that for any gx which is a non-negative, bounded Lipschitz function, Eg S n,n E S n,n EgN asn. 2nlηn Hence, we obtain S, E S, d Eg EgN 2lη from the Toeplitz lemma. On the other hand, note that 6isequivalentto S, E S, d g EgN a.s. 2lη from Theorem 7. of Billingsley [3] and Section 2 of Peligrad and Shao [5]. Hence, to prove 6, it suffices to prove d g S, E S, Eg 2lη S, E S, a.s. 9 2lη for any gx which is a non-negative, bounded Lipschitz function. For any, let ξ g S, E S, S, E S, Eg. 2lη 2lη For any <,notethatg S, E S, 2lη andg S, E S, b i, X i E X i are independent and 2lη gx is a non-negative, bounded Lipschitz function. By the definition of η,weget Eξ ξ g Cov g Cov S, E S,, g 2lη S, E S,, g 2lη S, E S, 2lη S, E S, 2lη S, E S, g b i, X i E X i 2lη E b i, X i E X i E b i, X i E X i 2 b2 i, E X i 2 lη lη lη b i, + b +, 2 lη b2 i, + b2 +, lη + ln 2 / /4.
Wu and Chen Journal of Inequalities and Applications 23, 23:29 Page 8 of 2 http://www.ournalofinequalitiesandapplications.com/content/23//29 By Lemma 2.2,9holds. Now we prove 7. Let Z I Xi > η EI Xi > η for any. It is nown that IA B IB IA for any sets A and B. Thenfor <, by Lemma 2.ii and 5, we get P X > η o lη η 2 o. 2 Hence EZ Z Cov I Xi > η, I Xi > η Cov I Xi > η, I Xi > η I Xi > η i+ E I Xi > η I Xi > η i+ EI Xi > η P X > η. By Lemma 2.2,7holds. Finally, we prove 8. Let V 2 V 2 ζ f Ef lη lη for any. For <, Eζ ζ V f 2 Cov V 2, f lη lη f Cov V 2 V 2, f f lη lη V 2 X i 2 I X i η lη E X i 2 I X i η c EX 2 I X η c lη lη lη lη c. By Lemma 2.2,8 holds. This completes the proof of Lemma 2.3.
Wu and Chen Journal of Inequalities and Applications 23, 23:29 Page 9 of 2 http://www.ournalofinequalitiesandapplications.com/content/23//29 Proof of Theorem. Let U i S i /i; then 7isequivalentto d I 2V ln U i x x a.s.x R. 2 Let q 4/3, 2, then E X < and E X q < from Remar.4. Using the Marciniewicz-Zygmund strong large number law, we have U S a.s. S o /q a.s. Hence let a 2 ± εlη for any given < ε <,by ln + x x Ox 2 for x < /2, a ln U i a U i lη c lη U i 2 i 2/q 3/2 2/q lη a.s., from 3/2 2/q >,lx is a slowly varying function at,andη +. Therefore, for any δ >andalmosteveryeventω,thereexists ω, δ, x suchthat for >, I a U i x δ I a I a ln U i x Note that under the condition X η,, U i x + δ. 22 U i S i i i i i X i i X b, X S,. 23 Thus, by 22and23, for any given < ε <,δ >,wehavefor >, I 2V S, ln U i x I 2 + εlη x + δ + I V 2 >+εlη + I Xi > η for x,
Wu and Chen Journal of Inequalities and Applications 23, 23:29 Page of 2 http://www.ournalofinequalitiesandapplications.com/content/23//29 I 2V S, ln U i x I 2 εlη x + δ + I V 2 < εlη + I Xi > η for x <, and I 2V I 2V S, ln U i x I 2 εlη x δ I V 2 < εlη I Xi > η for x, S, ln U i x I 2 + εlη x δ I V 2 >+εlη I Xi > η for x <. Hence, to prove 2, it suffices to prove S, d I 2lη ± εx ± δ ± εx ± δ a.s., 24 d I Xi > η a.s., 25 d I V 2 >+εlη a.s., 26 d I V 2 < εlη a.s. 27 for any < ε <andδ >. Firstly, we prove 24. Let < β < /2, and let h be a real function such that for any given x R, Iy ± εx ± δ β hy Iy ± εx ± δ + β. 28 By EX i, Lemma 2.iii and 5, we have E S, E b i, X i I X i η i o lη. b i, E X i I X i > η E X I X > η olη η
Wu and Chen Journal of Inequalities and Applications 23, 23:29 Page of 2 http://www.ournalofinequalitiesandapplications.com/content/23//29 This, combining with 6, 28 and the arbitrariness of β in 28, 24, holds. By 7, 2 and the Toeplitz lemma, d I Xi > η d P X > η a.s. d EI Xi > η Hence, 25holds. Now we prove 26. For any λ >,letf be a non-negative, bounded Lipschitz function such that Ix >+λ f x Ix >+λ/2. From E V 2 lη, X ni is i.i.d., Lemma 2.iv, and 5, P V 2 > + λ lη P V 2 2 E V 2 > λlη /2 E V 2 E V 2 2 2 l 2 η oη2 o. lη Therefore, from 8and the Toeplitz lemma, d I V 2 >+λlη V 2 d Ef lη EX 4 I X η l 2 η V 2 d f lη d EI V 2 >+λ/2lη d P V 2 >+λ/2lη a.s. Hence, 26 holds. By similar methods used to prove 26, we can prove 27. This completes the proof of Theorem.. Competing interests The authors declare that they have no competing interests. Authors contributions QW conceived of the study and drafted, complete the manuscript. PC participated in the discussion of the manuscript. QW and PC read and approved the final manuscript. Author details College of Science, Guilin University of Technology, Guilin, 544, P.R. China. 2 Department of Mathematics, Ji nan University, Guangzhou, 563, P.R. China.
Wu and Chen Journal of Inequalities and Applications 23, 23:29 Page 2 of 2 http://www.ournalofinequalitiesandapplications.com/content/23//29 Authors information Qunying Wu, Professor, Doctor, woring in the field of probability and statistics. Acnowledgements The authors are very grateful to the referees and the editors for their valuable comments and some helpful suggestions that improved the clarity and readability of the paper. Supported by the National Natural Science Foundation of China 62, and the support Program of the Guangxi China Science Foundation 22GXNSFAA53. Received: 6 November 22 Accepted: 5 March 23 Published: 27 March 23 References. Arnold, BC, Villaseñor, JA: The asymptotic distribution of sums of records. Extremes 3, 35-363 998 2. Beres, I, Csái, E: A universal result in almost sure central it theory. Stoch. Process. Appl. 94, 5-34 2 3. Billingsley, P: Convergence of Probability Measures. Wiley, New Yor 968 4. Brosamler, GA: An almost everywhere central it theorem. Math. Proc. Camb. Philos. Soc. 4, 56-574 988 5. Chandraseharan, K, Minashisundaram, S: Typical Means. Oxford University Press, Oxford 952 6. Csörgo, M, Szyszowicz, B, Wang, QY: Donser s theorem for self-normalized partial sums processes. Ann. Probab. 33, 228-24 23 7. Gine, E, Götze, F, Mason, DM: When is the Student t-statistic asymptotically standard normal? Ann. Probab. 25, 54-53 997 8. Gonchigdanzan, K, Rempala, G: A note on the almost sure central it theorem for the product of partial sums. Appl. Math. Lett. 9, 9-96 26 9. Hörmann, S: Critical behavior in almost sure central it theory. J. Theor. Probab. 2, 63-636 27. Huang, SH, Pang, TX: An almost sure central it theorem for self-normalized partial sums. Comput. Math. Appl. 6, 2639-2644 2. Ibragimov, IA, Lifshits, M: On the convergence of generalized moments in almost sure central it theorem. Stat. Probab. Lett. 4,343-35 998 2. Lacey, MT, Philipp, W: A note on the almost sure central it theorem. Stat. Probab. Lett. 9, 2-25 99 3. Miao, Y: Central it theorem and almost sure central it theorem for the product of some partial sums. Proc. Indian Acad. Sci. Math. Sci. 82, 289-294 28 4. Pang, TX, Lin, ZY, Hwang, KS: Asymptotics for self-normalized random products of sums of i.i.d. random variables. J. Math. Anal. Appl. 334, 246-259 27 5. Peligrad, M, Shao, QM: A note on the almost sure central it theorem for wealy dependent random variables. Stat. Probab. Lett. 22,3-36 995 6. Rempala, G, Wesolowsi, J: Asymptotics for products of sums and U-statistics. Electron. Commun. Probab. 7, 47-54 22 7. Schatte, P: On strong versions of the central it theorem. Math. Nachr. 37,249-256 988 8. Wu, QY: Almost sure it theorems for stable distribution. Stat. Probab. Lett. 86, 662-672 2 9. Wu, QY: An almost sure central it theorem for the weight function sequences of NA random variables. Proc. Indian Acad. Sci. Math. Sci. 23, 369-377 2 2. Wu, QY: Almost sure central it theory for products of sums of partial sums. Appl. Math. J. Chin. Univ. Ser. B 272, 69-8 22 2. Wu, QY: A note on the almost sure it theorem for self-normalized partial sums of random variables in the domain of attraction of the normal law. J. Inequal. Appl. 22, 7 22..86/29-242X-22-7 22. Ye, DX, Wu, QY: Almost sure central it theorem for product of partial sums of strongly mixing random variables. J. Inequal. Appl. 2,Article ID 5763 2 23. Zhang, Y, Yang, XY: An almost sure central it theorem for self-normalized products of sums of i.i.d. random variables. J. Math. Anal. Appl. 376, 29-4 2 doi:.86/29-242x-23-29 Cite this article as: Wu and Chen: An improved result in almost sure central it theorem for self-normalized products of partial sums. Journal of Inequalities and Applications 23 23:29.