Thông tin tài liệu
Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010, Article ID 569759, 14 pages doi:10.1155/2010/569759 Research Article AH´ajek-R´enyi-Type Maximal Inequality and Strong Laws of Large Numbers for Multidimensional Arrays Nguyen Van Quang 1 and Nguyen Van Huan 2 1 Department of Mathematics, Vinh University, Nghe An 42000, Vietnam 2 Department of Mathematics, Dong Thap University, Dong Thap 871000, Vietnam Correspondence should be addressed to Nguyen Van Huan, vanhuandhdt@yahoo.com Received 1 July 2010; Accepted 27 October 2010 Academic Editor: Alexander I. Domoshnitsky Copyright q 2010 N. V. Quang and N. Van Huan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. AH ´ ajek-R ´ enyi-type maximal inequality is established for multidimensional arrays of random elements. Using this result, we establish some strong laws of large numbers for multidimensional arrays. We also provide some characterizations of Banach spaces. 1. Introduction and Preliminaries Throughout this paper, the symbol C will denote a generic positive constant which is not necessarily the same one in each appearance. Let d be a positive integer, the set of all nonnegative integer d-dimensional lattice points will be denoted by d 0 ,andthesetofall positive integer d-dimensional lattice points will be denoted by d . We will write 1, m, n,and n 1 for points 1, 1, ,1, m 1 ,m 2 , ,m d , n 1 ,n 2 , ,n d ,andn 1 1,n 2 1, ,n d 1, respectively. The notation m n or n m means that m i n i for all i 1, 2, ,d,the limit n →∞is interpreted as n i →∞for a ll i 1, 2, ,d this limit is equivalent to min{n 1 ,n 2 , ,n d }→∞,andwedefine|n| d i1 n i . Let {b n , n ∈ d } be a d-dimensional array of real numbers. We define Δb n to be the dth-order finite difference of the b’s at the point n.Thus,b n 1kn Δb k for all n ∈ d .For example, if d 2, then for all i, j ∈ 2 , Δb ij b ij − b i,j−1 − b i−1,j b i−1,j−1 with the convention that b 0,0 b i,0 b 0,j 0. We say that {b n , n ∈ d } is a nondecreasing array if b k b l for any points k l. H ´ ajek and R ´ enyi 1 proved the following important inequality: If X j ,j 1 is a sequence of real-valued independent random variables with zero means and finite second 2 Journal of Inequalities and Applications moments, and b j ,j 1 is a nondecreasing sequence of positive real numbers, then for any ε>0 and for any positive integers n, n 0 n 0 <n, ⎛ ⎝ max n 0 i n 1 b i i j1 X j ε ⎞ ⎠ 1 ε 2 ⎛ ⎝ n 0 j1 X 2 j b 2 n 0 n jn 0 1 X 2 j b 2 j ⎞ ⎠ . 1.1 This inequality is a generalization of the Kolmogorov inequality and is a useful tool to prove the strong law of large numbers. Fazekas and Klesov 2 gave a general method for obtaining the strong law of lar ge numbers for sequences of random variables by using a H ´ ajek-R ´ enyi-type maximal inequality. Afterwards, Nosz ´ aly and T ´ om ´ acs 3 extended this result to multidimensional arrays see also Klesov et al. 4.Theyprovidedasufficient condition for d-dimensional arrays of random variables to satisfy the strong law of large numbers 1 b n 1kn X k −→ 0a.s.asn −→ ∞ , 1.2 where {b n , n ∈ d } is a positive, nondecreasing d-sequence of product type, that is, b n d i1 b i n i ,where{b i n i ,n i 1} is a nondecreasing sequence of positive real numbers for each i 1 , 2, ,d. Then, we have b n 1kn Δb k b 1 n 1 b 2 n 2 ···b d n d , n ∈ d . 1.3 This implies that Δb n b 1 n 1 − b 1 n 1 −1 b 2 n 2 − b 2 n 2 −1 ··· b d n d − b d n d −1 , n ∈ d . 1.4 Therefore, Δb n 0, n ∈ d , 1.5 Δb n Δb n1 Δb n 1 n 2 ···n d−1 ,n d 1 Δb n 1 1,n 2 1, ,n d−1 1,n d , n ∈ d . 1.6 On the other hand, we can show that under the assumption that {b n , n ∈ d } is an array of positive real numbers satisfying 1.5, it is not possible to guarantee tha t 1.6 holds for details, see Example 2.8 in the next section. Thus, if {b n , n ∈ d } is a positive, nondecreasing d-sequence of product type, then it is an array of positive real numbers satisfying 1.5, but the reverse is not true. In this paper, we use the hypothesis that {b n , n ∈ d } is an array of positive real numbers satisfying 1.5 and continue to study the problem of finding the sufficient condition for the strong law of large numbers 1.2. We also establish a H ´ ajek-R ´ enyi-type maximal inequality for multidimensional arrays of random elements and some maximal moment inequalities for arrays of dependent random elements. Journal of Inequalities and Applications 3 The paper is organized as follows. In the rest of this section, we recall some definitions and present some lemmas. Section 2 is devoted to our main results and their proofs. Let Ω, F, be a probability space. A family {F n , n ∈ d 0 } of nondecreasing sub-σ- algebras of F related to the partial order on d 0 is said to be a stochastic basic. Let {F n , n ∈ d 0 } be a stochastic basic such that F n {∅, Ω} if |n| 0, let E be a real separable Banach space, let BE be the σ-algebra of all Borel sets in E,andlet{X n , n ∈ d } be an array of random elements such that X n is F n /BE-measurable for all n ∈ d .Then {X n , F n , n ∈ d } is said to be an adapted array. For a given stochastic basic {F n , n ∈ d 0 },forn ∈ d 0 ,weset F 1 n k i 1 2 i d F n 1 k 2 k 3 ···k d : ∞ k 2 1 ∞ k 3 1 ··· ∞ k d 1 F n 1 k 2 k 3 ···k d , F j n k i 1 1 i j−1 k i 1 j1 i d F k 1 ···k j−1 n j k j1 ···k d if 1 <j<d, F d n k i 1 1 i d−1 F k 1 k 2 ···k d−1 n d , 1.7 in the case d 1, we set F 1 n F n . An adapted array {X n , F n , n ∈ d } is said to be a martingale difference array if X n |F i n−1 0foralln ∈ d and for all i 1, 2, ,d. In Quang and Huan 5, the authors showed that the set of all martingale difference arrays is really larger than the set of all arrays of independent mean zero random elements. A Banach space E is said to be p-uniformly smooth 1 p 2 if ρ τ sup x y x − y 2 − 1, ∀x, y ∈ E, x 1, y τ O τ p . 1.8 A Banach space E is said to be p-smoothable if there exists an equivalent norm under which E is p-uniformly smooth. Pisier 6 proved that a real separable Banach space E is p-smoothable 1 p 2 if and only if there exists a positive constant C such that for every L p integrable E-valued martingale difference sequence {X j , 1 j n}, n j1 X j p C n j1 X j p . 1.9 In Quang and Huan 5, this inequality was used to define p-uniformly smooth Banach spaces. Let {Y j ,j 1} be a sequence of independent identically distributed random variables with Y 1 1 Y 1 −11/2. Let E ∞ E × E × E ×··· and define E ⎧ ⎨ ⎩ v 1 ,v 2 , ∈ E ∞ : ∞ j1 Y j v j converges inprobability ⎫ ⎬ ⎭ . 1.10 4 Journal of Inequalities and Applications Let 1 p 2. Then, E is said to be of Rademacher type p if there exists a positive constant C such that ∞ j1 Y j v j p C ∞ j1 v j p ∀ v 1 ,v 2 , ∈ E . 1.11 It is well known that if a real separable Banach space is of Rademacher type p1 p 2, then it is of Rademacher type q for all 1 q p. Every real separable Banach space is of Rademacher type 1, while the L p -spaces and p -spaces are of Rademacher type 2∧p for p 1. The real line is of Rademacher type 2. Furthermore, if a Banach space is p-smoothable, then it is of Rademacher type p. For more details, the reader may refer to Borovskikh and Korolyuk 7, Pisier 8,andWoyczy ´ nski 9. Now, we present some lemmas which will be needed in what follows. The first lemma is a variation of Lemma 2.6 of Fazekas and T ´ om ´ acs 10 and is a multidimensional version of the Kronecker lemma. Lemma 1.1. Let {x n , n ∈ d } be an array of nonnegative real numbers, and let {b n , n ∈ d } be a nondecreasing array of positive real numbers such that b n →∞as n →∞.If n1 x n < ∞, 1.12 then 1 b n 1kn b k x k −→ 0 as n −→ ∞ . 1.13 Proof. For every ε>0, there exists a point n 0 ∈ d such that k1 x k − 1kn 0 x k ε. 1.14 Therefore, for a ll n n 0 , 0 1 b n 1kn b k x k − 1kn 0 b k x k 1kn x k − 1kn 0 x k ε. 1.15 It means that lim n →∞ 1 b n 1kn b k x k − 1kn 0 b k x k 0. 1.16 Journal of Inequalities and Applications 5 On the other hand, since b n →∞as n →∞, lim n →∞ 1 b n 1kn 0 b k x k 0. 1.17 Combining the above arguments, this completes the proof of Lemma 1.1. The proof of the next lemma is very simple and is therefore omitted. Lemma 1.2. Let Ω, F, be a probability space, and let {A n , n ∈ d } be an array of sets in F such that A n ⊂ A m for any points m n.Then, n1 A n lim n →∞ A n . 1.18 Lemma 1.3. Let {X n , n ∈ d } be an array of random elements. If for any ε>0, lim n →∞ sup kn X k ε 0, 1.19 then X n → 0 a.s. as n →∞. Proof. For each i 1, we have n1 kn X k 1 i lim n →∞ kn X k 1 i by Lemma 1.2 lim n →∞ sup kn X k 1 i 0. 1.20 Set A i 1 n1 kn X k 1 i . 1.21 Then, A0andforallω / ∈ A,foranyi 1, there exists a point l ∈ d such that X k ω < 1/i for all k l.Itmeansthat X k −→ 0 a.s. as k −→ ∞ . 1.22 The proof is completed. Lemma 1.4 Quang and Huan 5. Let 1 p 2,andletE be a real separable Banach space. Then, the following two statements are equivalent. 6 Journal of Inequalities and Applications i The Banach space E is p-smoothable. ii For every L p integrable martingale difference array {X n , F n , n ∈ d }, there exists a positive constant C p,d (depending only on p and d)suchthat 1kn X k p C p,d 1kn X k p , n ∈ d . 1.23 2. Main Results Theorem 2.1 provides a H ´ ajek-R ´ enyi-type maximal inequality for multidimensional arrays of random elements. This theorem is inspired by the work of Shorack and Smythe 11. Theorem 2.1. Let p>0,let{b n , n ∈ d } be an array of positive real numbers satisfying 1.5,and let {X n , n ∈ d } be an array of random elements in a real separable Banach space. Then, there exists a positive constant C p,d such that for any ε>0 and for any points m n, max mkn 1 b k 1lk X l ε C p,d ε p max 1kn 1lk X l b l b m p . 2.1 Proof. Since {b n , n ∈ d } is a nondecreasing array of positive real numbers, max mkn 1 b k 1lk X l ε max mkn 1 b k b m 1lk X l ε 2 max 1kn 1 b k b m 1lk X l ε 2 . 2.2 For k ∈ d ,set r k b k b m ,D k 1lk X l r l . 2.3 Then, by interchanging the order of summation, we obtain the following 1lk X l 1lk 1tl Δr t X l r l 1tk Δr t tlk X l r l . 2.4 Thus, since Δr t 0, max 1kn 1 r k 1lk X l 2 d max 1ln D l . 2.5 Journal of Inequalities and Applications 7 By 2.2 and 2.5 and the Markov inequality, we have max mkn 1 b k 1lk X l ε max 1ln D l ε 2 d1 2 pd1 ε p max 1ln D l p . 2.6 This completes the proof of the theorem. Now, we use Theorem 2.1 to prove a strong law of large numbers for multidimensional arrays of random elements. This result is inspired by Theorem 3.2 of Klesov et al. 4. Theorem 2.2. Let p>0,let{a n , n ∈ d } be an array of nonnegative real numbers, let {b n , n ∈ d } be an array of positive real numbers satisfying 1.5 and b n →∞as n →∞,andlet{X n , n ∈ d } be an array of random elements in a real separable Banach space such that for any points m n, max 1kn 1lk X l b l b m p C 1kn a k b k b m p . 2.7 Then, the condition n1 a n b p n < ∞ 2.8 implies 1.2. Proof. By 2.7 and Theorem 2.1,foranyε>0 and for any points m n,wehave max mkn 1 b k 1lk X l ε C ε p 1kn a k b k b m p . 2.9 This implies, by letting n →∞,that sup km 1 b k 1lk X l ε C ε p k1 a k b k b m p C ε p 1km a k b p m k1 a k b p k − 1km a k b p k . 2.10 8 Journal of Inequalities and Applications Letting m →∞,by2.8 and Lemma 1.1,weobtain lim m →∞ sup km 1 b k 1lk X l ε 0. 2.11 Lemma 1.3 ensures that 1.2 holds. The proof is completed. The next theorem provides three characterizations of p-smoothable Banach spaces. The equivalence of i and ii is an improvement of a result of Quang and Huan 5 stated as Lemma 1.4 above. Theorem 2.3. Let 1 p 2,andletE be a real separable Banach space. Then, the following four statements are equivalent. i The Banach space E is p-smoothable. ii For every L p integrable martingale difference array {X n , F n , n ∈ d }, there exists a positive constant C p,d such that max 1kn 1lk X l p C p,d 1kn X k p , n ∈ d . 2.12 iii For every L p integrable martingale difference array {X n , F n , n ∈ d }, for every array of positive real numbers {b n , n ∈ d } satisfying 1.5, for any ε>0, and for any points m n, there exists a positive constant C p,d such that max mkn 1 b k 1lk X l ε C p,d ε p 1kn X k b k b m p . 2.13 iv For every martingale difference array {X n , F n , n ∈ d }, for every array of positive real numbers {b n , n ∈ d } satisfying 1.5 and b n →∞as n →∞, the condition n1 X n p b p n < ∞ 2.14 implies 1.2. Proof. i⇒ii: We easily obtain 2.12 in the case p 1. Now, we consider the case 1 <p 2. By virtue of Lemma 1.4,itsuffices to show that max 1kn 1lk X l p p p − 1 pd 1kn X k p , n ∈ d . 2.15 First, we remark that for d 1, 2.15 follows from Doob’s inequality. We assume that 2.15 holds for d D − 1 1, we wish to show that it holds for d D. Journal of Inequalities and Applications 9 For k ∈ D ,weset S k 1lk X l ,Y k D max 1 k i n i 1 i D−1 S k . 2.16 Then, S k 1 k 2 ···k D−1 k D |F D k 1 k 2 ···k D−1 ,k D −1 S k 1 k 2 ···k D−1 ,k D −1 |F D k 1 k 2 ···k D−1 ,k D −1 ⎛ ⎝ 1 l i k i 1 i D−1 X l 1 l 2 ···l D−1 k D |F D k 1 k 2 ···k D−1 ,k D −1 ⎞ ⎠ S k 1 k 2 ···k D−1 ,k D −1 . 2.17 Therefore, Y k D |F D k 1 k 2 ···k D−1 ,k D −1 max 1 k i n i 1 i D−1 S k |F D k 1 k 2 ···k D−1 ,k D −1 max 1 k i n i 1 i D−1 S k |F D k 1 k 2 ···k D−1 ,k D −1 Y k D−1 . 2.18 It means that {Y k D , F D k 1 k 2 ···k D−1 k D ,k D 1} is a nonnegative submartingale. Applying Doob’s inequality, we obtain max 1kn S k p max 1 k D n D Y k D p p p − 1 p Y p n D p p − 1 p max 1 k i n i 1 i D−1 S k 1 k 2 ···k D−1 n D p . 2.19 We set X D−1 k 1 k 2 ···k D−1 n D k D 1 X k 1 k 2 ···k D−1 k D , F D−1 k 1 k 2 ···k D−1 ∞ k D 1 F k 1 k 2 ···k D−1 k D . 2.20 10 Journal of Inequalities and Applications Then we again have that {X D−1 k 1 k 2 ···k D−1 , F D−1 k 1 k 2 ···k D−1 , k 1 ,k 2 , ,k D−1 ∈ D−1 } is a martingale difference array. Therefore, by the inductive assumption, we obtain max 1 k i n i 1 i D−1 S k 1 k 2 ···k D−1 n D p max 1 k i n i 1 i D−1 1 l i k i 1 i D−1 X D−1 l 1 l 2 ···l D−1 p p p − 1 pD−1 1 l i n i 1 i D−1 X D−1 l 1 l 2 ···l D−1 p p p − 1 pD−1 S n 1 n 2 ···n D p . 2.21 Combining 2.19 and 2.21 yields that 2.15 holds for d D. ii ⇒ iii:let{X n , F n , n ∈ d } be an arbitrary L p integrable martingale difference array. Then, for all m ∈ d , {X n /b n b m , F n , n ∈ d } is also an L p integrable martingale difference array. Therefore, the assertion ii and Theorem 2.1 ensure that 2.13 holds. iii ⇒ iv: t he proof of this implication is similar to the proof of Theorem 2.2 and is therefore omitted. iv ⇒ i: for a given positive integer d, assume that iv holds. Let {X j , F j ,j 1} be an arbitrary martingale difference sequence such that ∞ j1 X j p j p < ∞. 2.22 For n ∈ d ,set X n X n 1 if n i 1 2 i d , X n 0 if there exists a positive integer i 2 i d such that n i > 1, F n F n 1 ,b n n 1 . 2.23 Then, {X n , F n , n ∈ d } is a martingale difference array, and {b n , n ∈ d } is an array of positive real numbers satisfying 1.5 and b n →∞as n →∞.Moreover,weseethat n1 X n p b p n ∞ n 1 1 X n 1 p n p 1 < ∞, 2.24 [...]... 436–449, 2002 3 C Nosz´ ly and T Tom´ cs, A general approach to strong laws of large numbers for fields of a ´ a random variables,” Annales Universitatis Scientiarum Budapestinensis de Rolando E¨ tv¨ s Nominatae o o Sectio Mathematica, vol 43, pp 61–78, 2001 4 O Klesov, I Fazekas, C Nosz´ ly, and T Tom´ cs, Strong laws of large numbers for sequences and a ´ a fields,” Theory of Stochastic Processes, vol... supported by the National Foundation for Science Technology Development, Vietnam NAFOSTED , no 101.02.32.09 References 1 J H´ jek and A R´ nyi, “Generalization of an inequality of Kolmogorov,” Acta Mathematica Academiae a e Scientiarum Hungaricae, vol 6, pp 281–283, 1955 2 I Fazekas and O Klesov, A general approach to the strong laws of large numbers, ” Theory of Probability and Its Applications, vol 45,... 2008 5 N V Quang and N V Huan, A characterization of p-uniformly smooth Banach spaces and weak laws of large numbers for d-dimensional adapted arrays,” The Indian Journal of Statistics, vol 72, pp 344–358, 2010 6 G Pisier, “Martingales with values in uniformly convex spaces,” Israel Journal of Mathematics, vol 20, no 3-4, pp 326–350, 1975 7 Y V Borovskikh and V S Korolyuk, Martingale Approximation, VSP,... Ngamkham, N D Tien, and A I Volodin, “Marcinkiewicz-Zygmund type law of large numbers for double arrays of random elements in Banach spaces,” Lobachevskii Journal of Mathematics, vol 30, no 4, pp 337–346, 2009 18 N V Quang and N V Huan, “On the strong law of large numbers and Lp -convergence for double arrays of random elements in p-uniformly smooth Banach spaces,” Statistics & Probability Letters, vol... 1 statements are equivalent 2, and let E be a real separable Banach space Then, the following four 12 Journal of Inequalities and Applications i The Banach space E is of Rademacher type p ii For every array of Lp integrable independent mean zero random elements {Xn , n ∈ there exists a positive constant C p,d such that 2.12 holds Æ d }, iii For every array of Lp integrable independent mean zero random... numbers and the central limit theorem in Banach spaces,” Annals of Probability, vol 4, no 4, pp 587–599, 1976 13 S Gan, “The H` jek-R` nyi inequality for Banach space valued martingales and the p smoothness of a e Banach spaces,” Statistics & Probability Letters, vol 32, no 3, pp 245–248, 1997 14 S Gan and D Qiu, “On the H´ jek-R´ nyi inequality, ” Wuhan University Journal of Natural Sciences, vol a. .. Fazekas and T Tom´ cs, Strong laws of large numbers for pairwise independent random variables ´ a with multidimensional indices,” Publicationes Mathematicae Debrecen, vol 53, no 1-2, pp 149–161, 1998 11 G R Shorack and R T Smythe, “Inequalities for |Sk |/bk where k ∈ N r ,” Proceedings of the American Mathematical Society, vol 54, pp 331–336, 1976 12 J Hoffmann-Jørgensen and G Pisier, “The law of large. .. Utrecht, The Netherlands, 1997 8 G Pisier, “Probabilistic methods in the geometry of Banach spaces,” in Probability and Analysis (Varenna, 1985), vol 1206 of Lecture Notes in Mathematics, pp 167–241, Springer, Berlin, Germany, 1986 9 W A Woyczynski, “On Marcinkiewicz-Zygmund laws of large numbers in Banach spaces and related ´ rates of convergence,” Probability and Mathematical Statistics, vol 1, no... Christofides and R J Serfling, Maximal inequalities for multidimensionally indexed submartingale arrays,” The Annals of Probability, vol 18, no 2, pp 630–641, 1990 16 F Moricz, U Stadtmuller, and M Thalmaier, Strong laws for blockwise M-dependent random ´ ¨ fields,” Journal of Theoretical Probability, vol 21, no 3, pp 660–671, 2008 14 Journal of Inequalities and Applications 17 L V Dung, T Ngamkham, N D... holds for every p > 1 and for every array of independent mean zero random elements in a real separable Banach space Therefore, by using the implication 2.1.1 ⇒ 2.1.2 of Theorem 2.1 of Hoffmann-Jørgensen and Pisier 12 and the same arguments as in the proof of Theorem 2.3, we get the following theorem which generalizes some results given by Christofides and Serfling 15 and Gan and Qiu 14 We omit its proof . H ´ ajek-R ´ enyi-type maximal inequality for multidimensional arrays of random elements and some maximal moment inequalities for arrays of dependent random elements. Journal of Inequalities and Applications. “Generalization of an inequality of Kolmogorov,” Acta Mathematica Academiae Scientiarum Hungaricae, vol. 6, pp. 281–283, 1955. 2 I. Fazekas and O. Klesov, A general approach to the strong laws of large numbers, ”. Theory of Probability and Its Applications, vol. 45, no. 3, pp. 436–449, 2002. 3 C. Nosz ´ aly and T. T ´ om ´ acs, A general approach to strong laws of large numbers for fields of random variables,”
Ngày đăng: 21/06/2014, 07:20
Xem thêm: Báo cáo hóa học: " Research Article A Hajek-Renyi-Type Maximal Inequality and ´ ´ Strong Laws of Large Numbers for " doc, Báo cáo hóa học: " Research Article A Hajek-Renyi-Type Maximal Inequality and ´ ´ Strong Laws of Large Numbers for " doc
