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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. Some exponential inequalities for acceptable random variables and complete convergence Journal of Inequalities and Applications 2011, 2011:142 doi:10.1186/1029-242X-2011-142 Aiting Shen (shenaiting1114@126.com) Shuhe Hu (hushuhe@263.net) Andrei Volodin (volodin@math.uregina.ca) Xuejun Wang (wxjahdx2000@126.com) ISSN 1029-242X Article type Research Submission date 6 July 2011 Acceptance date 22 December 2011 Publication date 22 December 2011 Article URL http://www.journalofinequalitiesandapplications.com/content/2011/1/ 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 © 2011 Shen et al. ; 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.
Some exponential inequalities for acceptable random variables and complete convergence Aiting Shen1 , Shuhe Hu1 , Andrei Volodin∗2 and Xuejun Wang1 1 School of Mathematical Science, Anhui University, Hefei 230039, China 2 Department of Mathematics and Statistics, University of Regina, Regina Saskatchewan S4S 0A2, Canada ∗ Corresponding author: volodin@math.uregina.ca Email addresses: AS: baret@sohu.com SH: hushuhe@263.net XW: wxjahdx2000@126.com December 14, 2011 Abstract 1
Some exponential inequalities for a sequence of acceptable random variables are obtained, such as Bernstein-type inequality, Hoeﬀding-type inequality. The Bernstein-type inequality for acceptable random variables generalizes and improves the corresponding results presented by Yang for NA random variables and Wang et al. for NOD random variables. Using the exponential inequalities, we further study the complete convergence for acceptable random variables. MSC(2000): 60E15, 60F15. Keywords: acceptable random variables; exponential inequality; complete conver- gence. 1 Introduction Let {Xn , n ≥ 1} be a sequence of random variables deﬁned on a ﬁxed probability space n (Ω, F , P ). The exponential inequality for the partial sums i=1 (Xi − EXi ) plays an important role in various proofs of limit theorems. In particular, it provides a measure of convergence rate for the strong law of large numbers. There exist several versions available in the literature for independent random variables with assumptions of uniform boundedness or some, quite relaxed, control on their moments. If the independent case is classical in the literature, the treatment of dependent variables is more recent. First, we will recall the deﬁnitions of some dependence structure. Deﬁnition 1.1. A ﬁnite collection of random variables X1 , X2 , . . . , Xn is said to be negatively associated (NA) if for every pair of disjoint subsets A1 , A2 of {1, 2, . . . , n}, Cov {f (Xi : i ∈ A1 ), g (Xj : j ∈ A2 )} ≤ 0, (1.1) whenever f and g are coordinatewise nondecreasing (or coordinatewise nonincreasing) such that this covariance exists. An inﬁnite sequence of random variables {Xn , n ≥ 1} is NA if every ﬁnite subcollection is NA. Deﬁnition 1.2. A ﬁnite collection of random variables X1 , X2 , . . . , Xn is said to be negatively upper orthant dependent (NUOD) if for all real numbers x1 , x2 , . . . , xn , n P (Xi > xi , i = 1, 2, . . . , n) ≤ P (Xi > xi ), (1.2) i=1 2
and negatively lower orthant dependent (NLOD) if for all real numbers x1 , x2 , . . . , xn , n P (Xi ≤ xi , i = 1, 2, . . . , n) ≤ P (Xi ≤ xi ). (1.3) i=1 A ﬁnite collection of random variables X1 , X2 , . . . , Xn is said to be negatively orthant dependent (NOD) if they are both NUOD and NLOD. An inﬁnite sequence {Xn , n ≥ 1} is said to be NOD if every ﬁnite subcollection is NOD. The concept of NA random variables was introduced by Alam and Saxena [1] and carefully studied by Joag-Dev and Proschan [2]. Joag-Dev and Proschan [2] pointed out that a number of well-known multivariate distributions possesses the negative association property, such as multinomial, convolution of unlike multinomial, multivariate hyperge- ometric, Dirichlet, permutation distribution, negatively correlated normal distribution, random sampling without replacement, and joint distribution of ranks. The notion of NOD random variables was introduced by Lehmann [3] and developed in Joag-Dev and Proschan [2]. Obviously, independent random variables are NOD. Joag-Dev and Proschan [2] pointed out that NA random variables are NOD, but neither NUOD nor NLOD implies NA. They also presented an example in which X = (X1 , X2 , X3 , X4 ) possesses NOD, but does not possess NA. Hence, we can see that NOD is weaker than NA. Recently, Giuliano et al. [4] introduced the following notion of acceptability. Deﬁnition 1.3. We say that a ﬁnite collection of random variables X1 , X2 , . . . , Xn is acceptable if for any real λ, n n E exp λ Xi ≤ E exp(λXi ). (1.4) i=1 i=1 An inﬁnite sequence of random variables {Xn , n ≥ 1} is acceptable if every ﬁnite subcol- lection is acceptable. Since it is required that the inequality (1.4) holds for all λ, Sung et al. [5] weakened the condition on λ and gave the following deﬁnition of acceptability. Deﬁnition 1.4. We say that a ﬁnite collection of random variables X1 , X2 , . . . , Xn is acceptable if there exists δ > 0 such that for any real λ ∈ (−δ, δ ), n n E exp λ Xi ≤ E exp(λXi ). (1.5) i=1 i=1 An inﬁnite sequence of random variables {Xn , n ≥ 1} is acceptable if every ﬁnite subcol- lection is acceptable. 3
First, we point out that Deﬁnition 1.3 of acceptability will be used in the current article. As is mentioned in Giuliano et al. [4], a sequence of NOD random variables with a ﬁnite Laplace transform or ﬁnite moment generating function near zero (and hence a sequence of NA random variables with ﬁnite Laplace transform, too) provides us an example of acceptable random variables. For example, Xing et al. [6] consider a strictly stationary NA sequence of random variables. According to the sentence above, a sequence of strictly stationary and NA random variables is acceptable. Another interesting example of a sequence {Zn , n ≥ 1} of acceptable random variables can be constructed in the following way. Feller [7, Problem III.1] (cf. also Romano and Siegel [8, Section 4.30]) provides an example of two random variables X and Y such that the density of their sum is the convolution of their densities, yet they are not independent. It is easy to see that X and Y are not negatively dependent either. Since they are bounded, their Laplace transforms E exp(λX ) and E exp(λY ) are ﬁnite for any λ. Next, since the density of their sum is the convolution of their densities, we have E exp(λ(X + Y )) = E exp(λX )E exp(λY ). The announced sequence of acceptable random variables {Zn , n ≥ 1} can be now con- structed in the following way. Let (Xk , Yk ) be independent copies of the random vector (X, Y ), k ≥ 1. For any n ≥ 1, set Zn = Xk if n = 2k + 1 and Zn = Yk if n = 2k . Hence, the model of acceptable random variables that we consider in this article (Deﬁnition 1.3) is more general than models considered in the previous literature. Studying the limiting behavior of acceptable random variables is of interest. Recently, Sung et al. [5] established an exponential inequality for a random variable with the ﬁnite Laplace transform. Using this inequality, they obtained an exponential inequality for identically distributed acceptable random variables which have the ﬁnite Laplace transforms. The main purpose of the article is to establish some exponential inequalities for acceptable random variables under very mild conditions. Furthermore, we will study the complete convergence for acceptable random variables using the exponential inequalities. Throughout the article, let {Xn , n ≥ 1} be a sequence of acceptable random variables and denote Sn = n=1 Xi for each n ≥ 1. i Remark 1.1. If {Xn , n ≥ 1} is a sequence of acceptable random variables, then {−Xn , n ≥ 4
1} is still a sequence of acceptable random variables. Furthermore, we have for each n ≥ 1, n n n E exp λ (Xi − EXi ) = exp −λ EXi E exp λ Xi i=1 i=1 i=1 n n ≤ exp (−λEXi ) E exp(λXi ) i=1 i=1 n = E exp(λ(Xi − EXi )). i=1 Hence, {Xn − EXn , n ≥ 1} is also a sequence of acceptable random variables. The following lemma is useful. Lemma 1.1. If X is a random variable such that a ≤ X ≤ b, where a and b are ﬁnite real numbers, then for any real number h, b − EX ha EX − a hb EehX ≤ e+ e. (1.6) b−a b−a Proof. Since the exponential function exp(hX ) is convex, its graph is bounded above on the interval a ≤ X ≤ b by the straight line which connects its ordinates at X = a and X = b. Thus ehb − eha b − X ha X − a hb ehX ≤ (X − a) + eha = e+ e, b−a b−a b−a which implies (1.6). The rest of the article is organized as follows. In Section 2, we will present some exponential inequalities for a sequence of acceptable random variables, such as Bernstein- type inequality, Hoeﬀding-type inequality. The Bernstein-type inequality for acceptable random variables generalizes and improves the corresponding results of Yang [9] for NA random variables and Wang et al. [10] for NOD random variables. In Section 3, we will study the complete convergence for acceptable random variables using the exponential inequalities established in Section 2. 2 Exponential inequalities for acceptable random vari- ables In this section, we will present some exponential inequalities for acceptable random vari- ables, such as Bernstein-type inequality and Hoeﬀding-type inequality. 5
Theorem 2.1. Let {Xn , n ≥ 1} be a sequence of acceptable random variables with EXi = 0 and EXi2 = σi < ∞ for each i ≥ 1. Denote Bn = n=1 σi for each n ≥ 1. If 2 2 2 i there exists a positive number c such that |Xi | ≤ cBn for each 1 ≤ i ≤ n, n ≥ 1, then for any ε > 0, 2 exp − ε2 1 − εc if εc ≤ 1, 2 P (Sn /Bn ≥ ε) ≤ (2.1) ε exp − 4c if εc ≥ 1. Proof. For ﬁxed n ≥ 1, take t > 0 such that tcBn ≤ 1. It is easily seen that |EXik | ≤ (cBn )k−2 EXi2 , k ≥ 2. Hence, ∞ tk t2 t2 t tXi EXik ≤ 1 + EXi2 1 + cBn + c2 Bn + · · · 2 Ee = 1+ k! 2 3 12 k=2 2 t2 t t t EXi2 1 + cBn EXi2 1 + cBn ≤ 1+ ≤ exp . 2 2 2 2 By Deﬁnition 1.3 and the inequality above, we have n n t2 2 t tSn tXi tXi Ee =E e ≤ Ee ≤ exp Bn 1 + cBn , 2 2 i=1 i=1 which implies that t2 2 t P (Sn /Bn ≥ ε) ≤ exp −tεBn + Bn 1 + cBn . (2.2) 2 2 ε 1 We take t = Bn when εc ≤ 1, and take t = cBn when εc > 1. Thus, the desired result (2.1) can be obtained immediately from (2.2). Theorem 2.2. Let {Xn , n ≥ 1} be a sequence of acceptable random variables with EXi = 0 and |Xi | ≤ b for each i ≥ 1, where b is a positive constant. Denote σi = EXi2 2 and Bn = n=1 σi for each n ≥ 1. Then, for any ε > 0, 2 2 i ε2 P (Sn ≥ ε) ≤ exp − (2.3) 2 2 2Bn + 3 bε and ε2 P (|Sn | ≥ ε) ≤ 2 exp − . (2.4) 2Bn + 2 bε 2 3 6
Proof. For any t > 0, by Taylor’s expansion, EXi = 0 and the inequality 1 + x ≤ ex , we can get that for i = 1, 2, . . . , n, ∞ ∞ E (tXi )j tj E |Xi |j E exp{tXi } = 1 + ≤ 1+ j! j! j =2 j =2 ∞ t2 σi 2 tj −2 E |Xi |j = 1+ (2.5) 12 2 σ j! 2i j =2 t2 σi 2 t2 σi 2 . = 1+ Fi (t) ≤ exp Fi (t) , 2 2 where ∞ tj −2 E |Xi |j Fi (t) = , i = 1, 2, . . . , n. 12 σ j! 2i j =2 bε Denote C = b/3 and Mn = + 1. Choosing t > 0 such that tC < 1 and 2 3Bn Mn − 1 Cε tC ≤ = . 2 Mn Cε + Bn It is easy to check that for i = 1, 2, . . . , n and j ≥ 2, 12 E |Xi |j ≤ σi bj −2 ≤ σi C j −2 j !, 2 2 which implies that for i = 1, 2, . . . , n, ∞ ∞ tj −2 E |Xi |j (tC )j −2 = (1 − tC )−1 ≤ Mn . Fi (t) = ≤ (2.6) 12 σ j! 2i j =2 j =2 By Markov’s inequality, Deﬁnition 1.3, (2.5) and (2.6), we can get n t2 Bn2 −tε −tε P (Sn ≥ ε) ≤ e E exp {tSn } ≤ e E exp{tXi } ≤ exp −tε + Mn . (2.7) 2 i=1 Taking t = B 2ε n = Cε+B 2 . It is easily seen that tC < 1 and tC = CεCε 2 . Substituting ε nM +Bn n t = B 2ε n into the right-hand side of (2.7), we can obtain (2.3) immediately. By (2.3), we nM have ε2 P (Sn ≤ −ε) = P (−Sn ≥ ε) ≤ exp − 2 2 , (2.8) 2Bn + 3 bε since {−Xn , n ≥ 1} is still a sequence of acceptable random variables. The desired result (2.4) follows from (2.3) and (2.8) immediately. 7
Remark 2.1. By Theorem 2.2, we can get that for any t > 0, n2 t2 P (|Sn | ≥ nt) ≤ 2 exp − 2 2 2Bn + 3 bnt and t2 P (|Sn | ≥ Bn t) ≤ 2 exp − . 2+ 2 · bt 3 Bn 22 It is well known that the upper bound of P (|Sn | ≥ nt) is also 2 exp − 2B 2 +t2 bnt . So n n3 Theorem 2.3 extends corresponding results for independent random variables without necessarily adding any extra conditions. In addition, it is easy to check that ε2 ε2 exp − < exp − , 2Bn + 2 bε 2 2(2Bn + bε) 2 3 which implies that our Theorem 2.2 generalizes and improves the corresponding results of Yang [9, Lemma 3.5] for NA random variables and Wang et al. [10, Theorem 2.3] for NOD random variables. In the following, we will provide the Hoeﬀding-type inequality for acceptable random variables. Theorem 2.3. Let {Xn , n ≥ 1} be a sequence of acceptable random variables. If there exist two sequences of real numbers {an , n ≥ 1} and {bn , n ≥ 1} such that ai ≤ Xi ≤ bi for each i ≥ 1, then for any ε > 0 and n ≥ 1, 2ε2 P (Sn − ESn ≥ ε) ≤ exp − , (2.9) n 2 i=1 (bi − ai ) 2ε2 P (Sn − ESn ≤ −ε) ≤ exp − , (2.10) n − ai )2 i=1 (bi and 2ε2 P (|Sn − ESn | ≥ ε) ≤ 2 exp − . (2.11) n 2 i=1 (bi − ai ) Proof. For any h > 0, by Markov’s inequality, we can see that P (Sn − ESn ≥ ε) ≤ Eeh(Sn −ESn −ε) . (2.12) It follows from Remark 1.1 that n n h(Sn −ESn −ε) −hε h(Xi −EXi ) −hε Eeh(Xi −EXi ) . Ee =e E e ≤e (2.13) i=1 i=1 8
Denote EXi = µi for each i ≥ 1. By ai ≤ Xi ≤ bi and Lemma 1.1, we have bi − µi hai µi − ai hbi . Eeh(Xi −EXi ) ≤ e−hµi = eL(hi ) , e+ e (2.14) bi − ai bi − ai where µi − ai L(hi ) = −hi pi + ln(1 − pi + pi ehi ), hi = h(bi − ai ), pi = . bi − ai The ﬁrst two derivatives of L(hi ) with respect to hi are pi (1 − pi )e−hi pi L (hi ) = −pi + , L (hi ) = . (2.15) [(1 − pi )e−hi + pi ]2 (1 − pi )e−hi + pi The last ratio is of the form u(1 − u), where 0 < u < 1. Hence, (1 − pi )e−hi (1 − pi )e−hi 1 L (hi ) = 1− ≤. (2.16) (1 − pi )e−hi + pi (1 − pi )e−hi + pi 4 Therefore, by Taylor’s expansion and (2.16), we can get 1 1 1 L(hi ) ≤ L(0) + L (0)hi + h2 = h2 = h2 (bi − ai )2 . (2.17) i i 8 8 8 By (2.12), (2.13), and (2.17), we have n 1 P (Sn − ESn ≥ ε) ≤ exp −hε + h2 (bi − ai )2 . (2.18) 8 i=1 It is easily seen that the right-hand side of (2.18) has its minimum at h = n (4bεi −ai )2 . i=1 Inserting this value in (2.18), we can obtain (2.9) immediately. Since {−Xn , n ≥ 1} is a sequence of acceptable random variables, (2.9) implies (2.10). Therefore, (2.11) follows from (2.9) and (2.10) immediately. This completes the proof of the theorem. 3 Complete convergence for acceptable random vari- ables In this section, we will present some complete convergence for a sequence of acceptable random variables. The concept of complete convergence was introduced by Hsu and Robbins [11] as follows. A sequence of random variables {Un , n ≥ 1} is said to converge completely to a constant C if ∞ P (|Un − C | > ε) < ∞ for all ε > 0. In view of the n=1 Borel–Cantelli lemma, this implies that Un → C almost surely (a.s.). The converse is true if the {Un , n ≥ 1} are independent. Hsu and Robbins [11] proved that the sequence 9
of arithmetic means of independent and identically distributed (i.i.d.) random variables converges completely to the expected value if the variance of the summands is ﬁnite. Erd¨s o [12] proved the converse. The result of Hsu–Robbins–Erd¨s is a fundamental theorem in o probability theory and has been generalized and extended in several directions by many authors. Deﬁne the space of sequences ∞ hbn < ∞ for every 0 < h < 1 . H= {bn } : n=1 The following results are based on the space of sequences H. Theorem 3.1. Let {Xn , n ≥ 1} be a sequence of acceptable random variables with EXi = 0 and |Xi | ≤ b for each i ≥ 1, where b is a positive constant. Assume that n 2 i=1 EXi = O (bn ) for some {bn } ∈ H. Then, b−1 Sn → 0 completely as n → ∞. (3.1) n Proof. For any ε > 0, it follows from Theorem 2.2 that ∞ ∞ b2 ε2 n P (|Sn | ≥ bn ε) ≤ 2 exp − n EXi2 + 2 bbn ε 2 i=1 3 n=1 n=1 ∞ ≤2 exp{−Cbn } < ∞, n=1 which implies (3.1). Here, C is a positive number not depending on n. Theorem 3.2. Let {Xn , n ≥ 1} be a sequence of acceptable random variables with |Xi | ≤ c < ∞ for each i ≥ 1, where c is a positive constant. Then, for every {bn } ∈ H, (bn n)−1/2 (Sn − ESn ) → 0 completely as n → ∞. (3.2) Proof. For any ε > 0, it follows from Theorem 2.3 that ∞ ∞ bn ε2 1/2 P |Sn − ESn | ≥ (bn n) ε ≤2 exp − 2 < ∞, 2c n=1 n=1 which implies (3.2). 10
Theorem 3.3. Let {Xn , n ≥ 1} be a sequence of acceptable random variables with EXi = 0 and EXi2 = σi < ∞ for each i ≥ 1. Denote Bn = n=1 σi for each n ≥ 1. For 2 2 2 i ﬁxed n ≥ 1, there exists a positive number H such that m! 2 m−2 |EXim | ≤ σH , i = 1, 2, . . . , n (3.3) 2i for any positive integer m ≥ 2. Then, b−1 Sn → 0 completely as n → ∞, (3.4) n provided that {b2 /Bn } ∈ H and {bn } ∈ H. 2 n Proof. By (3.3), we can see that t2 2 t3 t2 2 EetXi = 1 + σi + EXi3 + · · · ≤ 1 + σi 1 + H |t| + H 2 t2 + · · · 2 6 2 1 for i = 1, 2, . . . , n, n ≥ 1. When |t| ≤ , it follows that 2H t2 σi 2 1 22 EetXi ≤ 1 + ≤ 1 + t2 σi ≤ et σi , 2 · i = 1, 2, . . . , n. (3.5) 2 1 − H |t | Therefore, by Markov’s inequality, Deﬁnition 1.3 and (3.5), we can get that for any x ≥ 0 and |t| ≤ 21 , H n n n P Xi ≥ x =P Xi ≥ x +P (−Xi ) ≥ x i=1 i=1 i=1 n n ≤ e−|t|x E exp |t| + e−|t|x E exp |t| Xi (−Xi ) i=1 i=1 n n ≤ e−tx E exp |t| + e−tx E exp |t| Xi (−Xi ) i=1 i=1 n n = e−tx E exp t + e−tx E exp t Xi (−Xi ) i=1 i=1 n n ≤ e−tx EetXi + Ee−tXi i=1 i=1 2 2 ≤ 2 exp −tx + t Bn . Hence, n ≤ 2 min exp −tx + t2 Bn . 2 P Xi ≥ x 1 |t|≤ 2H i=1 11
2 Bn If 0 ≤ x ≤ , then H x2 2 x2 x min exp −tx + t2 Bn = exp − 2 x+ B = exp − ; 4Bn n 2 4 2 2 Bn 4 Bn 1 |t|≤ 2H 2 Bn if x ≥ , then H 1 1 x min exp −tx + t2 Bn = exp − 2 B2 x+ ≤ exp − . 4H 2 n 2H 4H 1 |t|≤ 2H From the statements above, we can get that x2 n − 2 2e 4Bn , 0 ≤ x ≤ Bn , 2 P Xi ≥ x ≤ H 2 x x ≥ Bn , 2e− 4H , i=1 H which implies that for any x ≥ 0, n x2 x P Xi ≥ x ≤ 2 exp − + 2 exp − . 2 4Bn 4H i=1 Therefore, the assumptions of {bn } yield that ∞ n ∞ ∞ b2 ε2 1 bn ε n P Xi ≥ ε ≤2 exp − +2 exp − < ∞. 2 bn 4Bn 4H n=1 n=1 n=1 i=1 This completes the proof of the theorem. Competing interests The authors declare that they have no competing interests. Authors’contributions Some exponential inequalities for a sequence of acceptable random variables are obtained, such as Bernstein-type inequality, Hoeﬀding-type inequality. The complete convergence is further studied by using the exponential inequalities. All authors read and approved the ﬁnal manuscript. 12
Acknowledgments The authors are most grateful to the editor and anonymous referee for the careful reading of the manuscript and valuable suggestions which helped in signiﬁcantly improving an earlier version of this article. The study was supported by the National Natural Science Foundation of China (11171001, 71071002, 11126176) and the Academic Innovation Team of Anhui University (KJTD001B). References [1] Alam, K, Saxena, KML: Positive dependence in multivariate distributions. Commun. Stat. Theory Methods 10, 1183–1196 (1981) [2] Joag-Dev, K, Proschan, F: Negative association of random variables with applications. Ann. Stat. 11(1), 286–295 (1983) [3] Lehmann, E: Some concepts of dependence. Ann. Math. Stat. 37, 1137–1153 (1966) [4] Giuliano, AR, Kozachenko, Y, Volodin, A: Convergence of series of dependent ϕ-subGaussian random variables. J. Math. Anal. Appl. 338, 1188–1203 (2008) [5] Sung, SH, Srisuradetchai, P, Volodin, A: A note on the exponential inequality for a class of dependent random variables. J. Korean Stat. Soc. 40(1), 109–114 (2011) [6] Xing, G, Yang, S, Liu, A, Wang, X: A remark on the exponential inequality for negatively associated random variables. J. Korean Stat. Soc. 38, 53–57 (2009) [7] Feller, W: An Introduction to Probability Theory and its Applications, vol. II, 2nd edn. Wiley, New York (1971) [8] Romano, JP, Siegel, AF: The Wadsworth & Brooks/Cole Statistics/Probability Series, Counterexam- ples in Probability and Statistics. Wadsworth & Brooks/Cole Advanced Books & Software, Monterey (1986) [9] Yang, SC: Uniformly asymptotic normality of the regression weighted estimator for negatively asso- ciated samples. Stat. Probab. Lett. 62, 101–110 (2003) [10] Wang, XJ, Hu, SH, Yang, WZ, Ling, NX: Exponential inequalities and inverse moment for NOD sequence. Stat. Probab. Lett. 80, 452–461 (2010) [11] Hsu, PL, Robbins, H: Complete convergence and the law of large numbers. Proc. Natl. Acad. Sci. USA 33(2), 25–31 (1947) [12] Erd¨s, P: On a theorem of Hsu and Robbins. Ann. Math. Stat. 20(2), 286–291 (1949) o 13