THÔNG TIN TÀI LIỆU
MINISTRY OF EDUCATION AND TRAINING VINH UNIVERSITY -* - NGUYEN THI THANH HIEN CONVERGENCE FOR SUMS OF DEPENDENT RANDOM ELEMENTS IN HILBERT SPACES Speciality: Theory of Probability and Mathematical Statistics Code: 9460106 A SUMMARY OF MATHEMATICS DOCTORAL THESIS NGHE AN - 2020 This work is completed at Vinh University Supervisor: Assoc Prof Dr Le Van Thanh Reviewer 1: Reviewer 2: Reviewer 3: The thesis will be defended at the university-level thesis evaluating council at Vinh University at , , The thesis can be found at: - Vietnam National Library - Nguyen Thuc Hao Library and Information Center - Vinh University PREFACE Rationale 1.1 The law of large numbers is a classical problem of probability theory, it asserts that the additive mean of independent random variables distributions converges in a certain sense on the expectation of random variables However, large numbers of mathematicians continue to be interested in studying mathematics and have many applications in statistics, econometrics, natural sciences and many other fields Therefore, the study of numerical law is not only theoretical but also practical 1.2 When we study about probability theorem, the independence of random variables is important However, random phenomena that occur in practice often depend on each other So, we have to study different types of dependencies of random variables to fit practical such as local dependence, negative association, negative dependence 1.3 The development of limit theorems in probability theory has led to more general results than classical results One of the general directions is that of the results obtained for the real-valued random variables that are given to the values in the Hilbert space When studying about limit theorems, many authors have obtained interesting results, such as Gilles Pisier, Michel Talagrand, Andrew Rosalsky, Pedro Teran, Nguyen Van Quang 1.4 The convergence of weighted sums of random variables has many important applications in stochastic control and mathematical statistics , these are the nonparametric multiple regression model and the least squares estimators With the above reasons, we have chosen the topic for the thesis as follows: “ Convergence for sums of dependent random elements in Hilbert spaces ” Objective of the research In this thesis, we establish the strong law of large numbers, the weak law of large numbers and complete convergence for sequences of dependent random elements in Hilbert spaces Subject of the research The research object of the thesis is: - Negatively associated random elements, negative dependence and pairwise negative de- pendence for random elements in Hilbert spaces; - Some limit theorems Scope of the research The thesis focuses on the study the dependence in probability theory, the convergence for sums of random variables Methodology of the research - We use the independent method of document study and we analyze the existing results, then develop them into models with similar structures, or more general models - We establish seminar groups under the guidance of instructors, and exchange with local and foreign scientists Contributions of the thesis The results of the thesis contribute to enriching the research direction of the law of large numbers and complete convergence for sequences of dependent random elements in Hilbert spaces The thesis can be used as a reference for students, masters students and PhD students belong to the specialty: Theory of Probability and Mathematical Statistics Overview and organization of the research 7.1 Overview of the research The concepts of pairwise negative dependence, negative dependence and negative association were introduced by Lehmann, by Ebrahimi and Ghosh and by Joag-Dev and Prochan Joag-Dev and Proschan pointed out that many useful distributions enjoy the negative association properties (and therefore they are negatively dependent) including multinomial distribution, multivariate hypergeometric distribution, Dirichlet distribution, and distribution of random sampling without replacement Because of its wide applications in multivariate statistical analysis and reliability, the notion of negative association has received considerable attention recently We refer the reader to Joag-Dev and Prochan for fundamental properties, Newman for the central limit theorem, Matula for the three series theorem, Shao for the Rosenthal type maximal inequality and the Kolmogorov exponential inequality The concept of association, a dependence structure stronger than pairwise negative dependence, was first extended to Hilbert space valued random elements by Burton, Dabrowski and Dehling in 1986 Ko, Kim and Han introduced the concept of negative association for random elements with values in real separable Hilbert spaces In 2014, Huan, Quang and Thuan present another concept of coordinatewise negative association for H-valued random elements which is more general than the concept of Ko, Kim and Han In many papers one can find some interesting results concerning sequences of H-valued negatively associated radom variables We refer only some of them Almost sure convergence by Ko, Kim and Han; almost sure convergence extending the results of Ko, Kim and Han by Thanh; the Hajek-Renyi inequality by Miao; the Baum-Katz type theorem by Huan, Quang and Thuan 7.2 The organization of the research Besides the sections of usual notations, preface, general conclusions, and recommendations, list of the author’s articles related to the thesis and references, the thesis is organized into three chapters Chapter presents some preliminaries Chapter studies some limit theorems such as laws of large numbers and complete convergence for sequences of pairwise and coordinatewise negative dependence for random elements in Hilbert spaces Chapter studies some limit theorems such as laws of large numbers and complete convergence for sequences of coordinatewise negatively associated for random elements in Hilbert spaces 4 CHAPTER PRELIMINARIES In this chapter, we introduce some basic knowledge of negatively dependent random variables, pairwise negatively dependent random variables, negatively associated random variables on a probability space (Ω; F; P) Then, we introduce some basic knowledge of coordinatewise negatively dependent random elements, pairwise and coordinatewise negatively dependent random elements, coordinatewise negatively associated random elements in Hilbert spaces Finally, we introduce some basic knowledge of the regularly varying function and slowly varying function 1.1 The negatively dependent random variables, the negatively associated random variables 1.1.1 Definition A collection of random variables {X1 , X2 , , Xn } is said to be i) negatively lower dependent, if for all x1 , , xn ∈ R, we have P(X1 ≤ x1 , , Xn ≤ xn ) ≤ P(X1 ≤ x1 ) P(Xn ≤ xn ), (1.1) ii) negatively upper dependent, if for all x1 , , xn ∈ R, we have P(X1 > x1 , , Xn > xn ) ≤ P(X1 > x1 ) P(Xn > xn ), (1.2) iii) negatively dependent, if they are both negatively lower dependent and negatively upper dependent A sequence of random variables {Xi , i ≥ 1} is said to be negatively dependent if for any n ≥ 1, the collection {X1 , , Xn } is negatively dependent 1.1.2 Definition A collection of random variables {X1 , , Xn } is said to be negatively associated if for any disjoint subsets A, B of {1, , n} and any real coordinatewise nondecreasing functions f on R|A| and g on R|B| , Cov(f (Xk , k ∈ A), g(Xk , k ∈ B)) ≤ (1.3) whenever the covariance exists, where |A| denotes the cardinality of A A sequence {Xn , n ≥ 1} of random variables is said to be negatively associated if every finite subfamily is negatively associated 1.2 The negatively associated random elements, the negatively dependent random elements 1.2.1 Definition Let H be a real separable Hilbert space with orthonormal basis {ej , j ∈ B} and inner product ·, · A sequence {Xi , i ≥ 1} of random elements with values in H is said to be negatively associated (NA) if for any d ≥ 1, the sequence of Rd -valued random elements {( Xi , e1 , , Xi , ed ), i ≥ 1} is negatively associated 1.2.2 Definition A sequence {Xi , i ≥ 1} of random elements taking values in H is said to be coordinatewise negatively associated (CNA) if for some orthonormal basis {ej , j ≥ 1} and for each j ≥ 1, the sequence of random variables { Xi , ej , i ≥ 1} is negatively associated In this section, we introduce the notion of coordinatewise negative dependence (CND), and pairwise and coordinatewise negative dependence (PCND) for random elements in Hilbert spaces 1.2.3 Definition A sequence {Xi , i ≥ 1} of random elements taking values in H is said to be coordinatewise negative dependence (CND) (resp., pairwise and coordinatewise negative dependence (PCND)) if for some orthonormal basis {ej , j ≥ 1} and for each j ≥ 1, the sequence of random variables { Xi , ej , i ≥ 1} is negatively dependent (resp., pairwise negatively dependent) The next, we prove the Rademacher-Menshov type inequality and the H´ajek-R´enyi type inequality for sums of PCND random elements in H 1.2.4 Theorem Let {Xn , n ≥ 1} be a sequence of PCND mean random elements in H satisfying E Xn < ∞ for all n ≥ Then for any n ≥ 1, we have n E Xi i=1 n E Xi , ≤ i=1 (1.4) and k 1≤k≤n n ≤ log2 (2n) Xi E max E Xi (1.5) i=1 i=1 1.2.5 Theorem Let {Xn , n ≥ 1} be a sequence of PCND mean random elements in H satisfying E Xn < ∞ for all n ≥ 1, and let {bn , n ≥ 1} be a sequence of positive nondecreasing constants Then for any n ≥ 1, k i=1 Xi E max bk 1≤k≤n n ≤ log (2n) i=1 E Xi b2i (1.6) Moreover, for any ≤ m ≤ n, E max m≤k≤n k X i i=1 bk ≤ m i=1 E b2m Xi n 2 + log (2(n − m)) i=m+1 E Xi b2i (1.7) 1.3 The slowly varying function 1.3.1 Definition A real-valued function R(·) is said to be regularly varying function with index of regular variation ρ (ρ ∈ R) if it is a positive and measurable function on [A, ∞) for some A > 0, and for each λ > 0, lim x→∞ R(λx) = λρ R(x) (1.8) A regularly varying function with the index of regular variation ρ = is called slowly varying function 1.3.2 Lemma Let p > and let L(·) be a slowly varying function defined on [A, ∞) for some A > 0, satisfying xL (x) = 0, x→∞ L(x) lim Then the following statements hold (i) There exists B ≥ A such that xp L(x) is increasing on [B, ∞), x−p L(x) is decreasing on [B, ∞) and limx→∞ xp L(x) = ∞, limx→∞ x−p L(x) = (iii) For all λ > 0, L(x) = x→∞ L(x + λ) lim Because of Lemma 1.3.2, we have 1.3.3 Lemma Let p > 1, q ∈ R, and let L(·) be a differentiable slowly varying function defined on [A, ∞) for some A > satisfying xL (x) = 0, x→∞ L(x) lim Then for n large enough, we have 2Lq (n) ≤ 3(p − 1)np−1 ∞ k=n Lq (k) (p + 1)Lq (n) ≤ kp (p − 1)np−1 (1.9) The following proposition gives a criterion for E (|X|α Lα (|X| + A)) < ∞ 1.3.4 Proposition Let α ≥ and let X be a random variable Let L(·) be a slowly varying function defined on [A, ∞), for some A > Assume that xα Lα (x) and x1/α L(x1/α ) are increasing on [A, ∞) Then E(|X|α Lα (|X| + A)) < ∞ if and only if P(|X| > bn ) < ∞ (1.10) n≥Aα where bn = n1/α L(n1/α ), n ≥ Aα The conclusions of Chapter In this chapter, we obtain some main results: - A brief some of the basic concepts and properties of negatively dependent random variables, pairwise negatively dependent random variables, negatively associated random variables; - A brief some of the basic concepts and properties of negatively associated random elements, coordinatewise negatively associated random elements in Hilbert spaces; - Present the new notion of coordinatewise negative dependence (CND), and pairwise and coordinatewise negative dependence (PCND) for random elements in Hilbert spaces; - Present and prove some classical inequalities of random elements in Hilbert spaces; - A brief some of the basic concepts and properties of regularly varying function, slowly varying function Present and prove some properties of regularly varying function, slowly varying function 8 CHAPTER THE LAW OF LARGE NUMBERS AND COMPLETE CONVERGENCE FOR SEQUENCES OF PAIRWISE AND COORDINATEWISE NEGATIVE DEPENDENCE FOR RANDOM ELEMENTS IN HILBERT SPACES In this chapter, we establish some limit theorems kind of law of large numbers and complete convergence for sequences of pairwise and coordinatewise negative dependence for random elements in Hilbert spaces 2.1 The strong law of large numbers and complete convergence In this section, we establish some strong laws of large numbers and complete convergence for sequences of pairwise and coordinatewise negative dependence for random elements in Hilbert spaces The following theorem is an extension of the classical the Rademacher-Menshov strong law of large numbers to the blockwise PCND random elements in Hilbert spaces 2.1.1 Theorem Let {Xn , n ≥ 1} be a sequence of mean random elements in H such that for any k ≥ 0, the random elements {Xi , 2k ≤ i < 2k+1 } are PCND Let {bn , n ≥ 1} be a nondecreasing sequence of positive constants satisfying b n+1 b2n+1 > and sup < ∞ n≥0 b2n n≥0 b2n inf If ∞ E Xn n=1 then lim n→∞ log2 n b2n n i=1 Xi bn < ∞, = a.s (2.1) (2.2) The next theorem in this section establishes the complete convergence for sequences of PCND elements in Hilbert spaces Let X be a random vector in H Here and thereafter, we denote the j th coordinate of X by X (j) , i.e., X (j) = X, ej , j ≥ Then, we can write X (j) ej X= j≥1 2.1.2 Theorem Let {Xn , n ≥ 1} be a sequence of PCND identically distributed mean random elements in H and ≤ p < 2, αp ≥ Let {ani , n ≥ 1, ≤ i ≤ n} is an array of constants satisfying n a2ni ≤ Kn, ∀n ≥ (2.3) E|X1(j) |p < ∞ (2.4) i=1 If j∈B then for all ε > 0, we have ∞ k αp−2 n P n=1 ani Xi > ε(n log2 n)α max 1≤k≤n < ∞ (2.5) i=1 In the special case, αp = and ani ≡ 1, we have a following corollary 2.1.3 Corollary Let {Xn , n ≥ 1} be a sequence of PCND identically distributed mean random elements in H, and let ≤ p < If (j) E|X1 |p < ∞, (2.6) j≥1 then for all ε > 0, we have ∞ k n n=1 −1 P Xi > ε(n log2 n)1/p max 1≤k≤n < ∞ (2.7) i=1 The following corollary is a Marcinkiewicz-Zygmund type strong law of large numbers for PCND random elements in Hilbert spaces 2.1.4 Corollary Let {Xn , n ≥ 1} be a sequence of PCND identically distributed mean random elements in H, and let ≤ p < If (j) E|X1 |p < ∞, j≥1 10 then n i=1 Xi (n log2 n)1/p → a.s as n → ∞ 2.2 The weak law of large numbers In the following theorem, we establish the weak law of large number for sequences of PCND elements in Hilbert spaces 2.2.1 Theorem Let {Xn , n ≥ 1} be a sequence of PCND random elements in H and {bn , n ≥ 1} be a sequence of positive constants For n ≥ 1, k ≥ 1, j ∈ B, we set (j) (j) (j) (j) (j) Ynk = −bn I Xk < −bn + Xk I |Xk | ≤ bn + bn I Xk > bn , (j) Ynk = Ynk ej j∈B If n P |Xk(j) | > bn = lim n→∞ and lim n→∞ bn (2.8) k=1 j∈B n E (j) Xk I |Xk(j) | ≤ bn =0 (2.9) k=1 j∈B then we obtain the weak law of large numbers bn n P (Xk − EYnk ) → 0, n → ∞ (2.10) k=1 The next, we establish the weak law of large number for sequences of PCND identically distributed random elements in Hilbert spaces 2.2.2 Theorem Let < p < 2, L(x) a differentiable slowly varying function on [A, ∞), for some A > satisfying xL (x) =0 x→∞ L(x) lim (2.11) and {ani , n ≥ 1, ≤ i ≤ n} is an array of constants satisfying n a2ni ≤ Kn, ∀n ≥ (2.12) i=1 Let {Xn , n ≥ 1} be a sequence of PCND identically distributed random elements in H For n ≥ 1, i ≥ 1, j ∈ B, we set bn = n1/p L(n + A), 11 (j) (j) Yni = −bn I Xi (j) (j) (j) < −bn + Xi I |Xi | ≤ bn + bn I Xi > bn , n (j) Yni ej , Sn Yni = Xi = i=1 j∈B If P |X1(j) | > bn = lim n n→∞ (2.13) j∈B and E(|X1(j) |2 I(|X1(j) | ≤ M )) < ∞, ∀M > (2.14) j∈B then we obtain the weak law of large numbers bn n P ani (Xi − EYni ) → n → ∞ (2.15) i=1 2.2.3 Corollary Let L(x) a differentiable slowly varying function on [A, ∞), for some A > satisfying xL (x) =0 x→∞ L(x) lim (2.16) and L(x) ≥ K on [A, ∞) Let {Xn , n ≥ 1} be a sequence of PCND identically distributed random elements in H satisfying (j) E|X1 | < ∞ EX1 = and (2.17) j∈B If {ani , n ≥ 1, ≤ i ≤ n} is an array of constants satisfying n a2ni ≤ Kn, ∀n ≥ (2.18) i=1 then we obtain the weak law of large numbers bn n P ani Xi → as n → ∞, (2.19) i=1 where bn = nL(n + A), n ≥ Conclusions of Chapter In Chapter of the thesis, some limit theorems such as laws of large numbers and the Baum - Katz type theorem on complete convergence for sequences of pairwise and coordinatewise negative dependent random elements are established To prove strong laws of large numbers 12 and the Baum - Katz type theorem, we often have to use the maximal type inequalities as a very important tool However, for the pairwise and coordinatewise negative dependent random elements, the Kolmogorov type inequality does not hold any more Instead, we have to prove the Rademacher-Menshov type inequality for this dependence struture To prove results on law of large numbers, we estimate the partial sums of random elements in each block [2k , 2k+1 ) and then apply a sub-sequences method This approach allows us to consider a sequence of blockwise, pairwise and coordinatewise negative dependent random elements For the weak laws of large numbers, we generalized the Feller weak law by considering the normalizing constants bn = nα L(n), where L(n) is a slowly varying function To achieve this result, we need to prove some properties of the slowly varying function, as shown in Lemma 1.3.6, Lemma 1.3.9 and Proposition 1.3.10 Besides, we also present some examples to illustrate the sharpness of the main theorems (Example 2.1.9, 2.2.2 and 2.2.3) 13 CHAPTER THE LAW OF LARGE NUMBERS AND COMPLETE CONVERGENCE FOR SEQUENCES OF COORDINATEWISE NEGATIVELY ASSOCIATED RANDOM ELEMENTS IN HILBERT SPACES In this chapter, we establish some limit theorems kind of law of large numbers and complete convergence for sequences of coordinatewise negatively associated random elements in Hilbert spaces 3.1 The strong law of large numbers and complete convergence In the following theorem, we establish complete convergence for weighted sums of coordinatewise negatively associated and identically distributed random elements in Hilbert spaces 3.1.1 Theorem Let ≤ p < 2, αp ≥ 1, {X, Xn , n ≥ 1} be a sequence of coordinatewise negatively associated and identically distributed random elements in H and let L(x) be a slowly varying function defined on [A, ∞) for some A > When p = 1, we assume further that L(x) ≥ and is increasing on [A, ∞) Let bn = nα L(nα ), n ≥ A1/α and {ani , n ≥ 1, ≤ i ≤ n} are constants satisfying n a2ni ≤ Kn, n ≥ (3.1) i=1 If random element X satisfies E |X (j) |p Lp (|X (j) | + A) < ∞, E(X) = 0, (3.2) j∈B then k n n≥A1/α αp−2 P max 1≤k≤n ani Xi > εbn < ∞ for all ε > 0, i=1 By applying Theorem 3.1.1, with ani ≡ and αp = 1, we obtain (3.3) 14 3.1.2 Corollary Let ≤ p < 2, {X, Xn , n ≥ 1} be a sequence of coordinatewise negatively associated and identically distributed random elements in H and let L(x) be a slowly varying function defined on [A, ∞) for some A > When p = 1, we assume further that L(x) ≥ and is increasing on [A, ∞) Let bn = n1/p L(n1/p ), n ≥ Ap If random element X satisfies E |X (j) |p Lp (|X (j) | + A) < ∞, E(X) = 0, (3.4) j∈B then k n n≥Ap −1 P Xi > εbn max 1≤k≤n < ∞ for all ε > 0, (3.5) i=1 The following corollary is the Marcinkiewicz-Zymund type strong law of large number for sequence of coordinatewise negatively associated and identically distributed random elements in Hilbert spaces 3.1.3 Corollary Let ≤ p < 2, {X, Xn , n ≥ 1} be a sequence of coordinatewise negatively associated and identically distributed random elements in H and let L(x) be a slowly varying function defined on [A, ∞) for some A > When p = 1, we assume further that L(x) ≥ and is increasing on [A, ∞) Let bn = n1/p L(n1/p ), n ≥ Ap If random element X satisfies E |X (j) |p Lp (|X (j) | + A) < ∞, E(X) = 0, (3.6) j∈B then lim max n→∞ bn 1≤k≤n k Xi = a.s (3.7) i=1 However, H is a finite dimensional Hilbert space We have the following theorem 3.1.4 Theorem Let ≤ p < 2, αp ≥ and {e1 , e2 , , ed } be an orthonormal basis in H {X, Xn , n ≥ 1} be a sequence of coordinatewise negatively associated and identically distributed random elements in H and let L(x) be a slowly varying function defined on [A, ∞) for some A > When p = 1, we assume further that L(x) ≥ and is increasing on [A, ∞) Let bn = nα L(nα ), n ≥ A1/α Then the following four statements are equivalent i) The random element X satisfies E(X) = 0, E ( X p p L ( X + A)) < ∞ (3.8) ii) For every array of constants {ani , n ≥ 1, ≤ i ≤ n} satisfying n a2ni ≤ Kn, ∀n ≥ 1, i=1 (3.9) 15 we have k αp−2 n P ani Xi > εbn max 1≤k≤n n≥A1/α < ∞ for all ε > (3.10) i=1 iii) k n αp−2 P Xi > εbn max 1≤k≤n n≥A1/α < ∞ for all ε > (3.11) i=1 iv) The strong law of large numbers k i=1 Xi max1≤k≤n lim bn n→∞ = a.s (3.12) holds 3.2 The weak law of large numbers For a sequence of random variables {Tn , n ≥ 1} and a sequence of positive constants {bn , n ≥ 1} we write bn = OP (bn ) to indicate that Tn /bn is bounded in probability; that is, lim sup P K→∞ n≥1 |Tn | >K bn = Let {Xn , n ≥ 1} be a sequence of random elements in H and {bn , n ≥ 1} be a sequence of positive constants For n ≥ 1, k ≥ 1, j ∈ B, we set (j) (j) (j) (j) (j) Ynk = −bn I(Xk < −bn ) + Xk I(|Xk | ≤ bn ) + bn I(Xk > bn ), (j) Ynk = Ynk ej j∈B The following theorem establishes the weak law of large numbers with random indices of CNA identically distributed random elements in Hilbert spaces 3.2.1 Theorem Let < p < and {Tn , n ≥ 1} is a sequence of positive integer-valued random variables satisfying Tn = OP (n), (3.13) and L(·) be a differentiable slowly varying function on [A, ∞) for some A > satisfying xL (x) = x→∞ L(x) lim (3.14) 16 Set bn = n1/p L(n + A), n ≥ Let {Xn , n ≥ 1} be a sequence of CNA, identically distributed satisfying nP (|X1 | > bn ) = 0, (j) (3.15) E |X1 |2 I(|X1 | ≤ M ) < ∞ for all M > (j) (3.16) lim n→∞ j∈B and (j) j∈B then we obtain the weak law of large numbers bn Tn P (Xk − EYnk ) → as n → ∞ (3.17) k=1 3.2.2 Corollary Let < p ≤ 1, L(·) be a differentiable slowly varying function on [A, ∞) for some A > satisfying xL (x) = x→∞ L(x) lim (3.18) and when p = 1, we asume further that L(x) ≥ K on [A, ∞) Set bn = n1/p L(n + A) Assume that {Xn , n ≥ 1} is a sequence of CNA, identically distributed random elements satisfying (j) E|X1 | < ∞ EX1 = and (3.19) j∈B If {Tn , n ≥ 1} is a sequence of positive integer-valued random variables satisfying Tn = OP (n), (3.20) then we obtain the weak law of large numbers bn Tn P Xk → as n → ∞ (3.21) k=1 The conclusions of Chapter Chapter of the thesis establishes some limit theorems such as laws of large numbers and the Baum - Katz type theorem for sequences of coordinatewise negatively associated random elements It is different from the coordinatewise and pairwise negative dependence structure, which studied in Chapter in the sense that for the coordinatewise negatively associated random elements, we are able to use the Kolmogorov type inequality Therefore, in comparation to the results presented in Chapter 2, the moment condition the assumptions are usually weaker, or the obtained conclusions are usually stronger if we put the same moment condition A main contribution of Chapter is the Baum - Katz theorem on complete convergence In this result, 17 we proved the Baum - Katz type theorem and the Marcinkiewicz - Zygmund type strong law of large numbers with very general normalization constants By considering coordinatewise negatively associated random elements taking values in finite dimensional Hilbert spaces, a very special case of Theorem 3.1.1 came close to a open problem raised by Chen and Sung in 2014 (see Theorem 3.1.5 and Corollary 3.1.9) Besides, we established weak laws of large numbers for the random sums in stead of determined sums which are considered in Chapter 18 GENERAL CONCLUSIONS AND SUGGESTIONS General conclusions In this thesis, we obtain some main results: - Introduce new notations, namely, the notion of coordinatewise negatively dependent random elements in Hilbert spaces and the notion of pairwise and coordinatewise negatively dependent random elements in Hilbert spaces; - Present and prove some properties of regularly varying function, slowly varying function; - Present and prove some classical inequalities of dependent random elements in Hilbert spaces and establish some limit theorems kind of law of large numbers for sequences of dependent random elements in Hilbert spaces; - Establish the Baum - Katz type theorem about the complete convergence for sequences of dependent random elements in Hilbert spaces; - Provide some examples to illustrate the theoretic results Recommendations In the near future we will study the following issues: - Establish some limit theorems kind of law of large numbers, complete convergence for sequences of other dependent random elements in Hilbert spaces; - Study some applications for weighted sums of dependent random elements; - Present the new notion of dependent random elements in other spaces; - Expand some classical inequalities with 2-index of dependent random elements in Hilbert spaces 19 LIST OF THE AUTHOR’S ARTICLES RELATED TO THE THESIS N T T Hien and L V Thanh (2015), On the weak laws of large numbers for sums of negatively associated random elements in Hilbert spaces, Statist Probab Lett., 107, 236–245 N T T Hien, L V Thanh and V T H Van (2019), On the negative dependence in Hilbert spaces with applications, Appl Math., 64, no 1, 45-59 V T N Anh, N T T Hien, L V Thanh and V T H Van (2019), The MarcinkiewiczZygmund type strong law of large numbers with general normalizing sequences, Journal of Theoretical Probability, Online first, https://link.springer.com/article/10.1007/s10959019-00973-2 V T N Anh and N T T Hien (2020), On the weak laws of large numbers for weighted sums of dependent identically distributed random elements in Hilbert spaces (submitted) Results of the thesis have been reported in: - The 5th National Conference: ”Probability- Statistics: Research, Application and teaching” (Da Nang University of Education, 23- 25/05/2015); - The 2th Mien Trung-Tay Nguyen Mathematics Conference (Da Lat University, 09- 11/12/2017); - Scientific workshop: “ Researching and teaching mathematics to meet the current requirements of educational innovation ” (Institute of Natural Education, Vinh University, September 19, 2019); - Seminar of Department of Probability - Statistics and Application, Institute of Science Education, Vinh University (from 2015 to 2019) ... negatively associated random elements in Hilbert spaces, Statist Probab Lett., 107, 236–245 N T T Hien, L V Thanh and V T H Van (2019), On the negative dependence in Hilbert spaces with applications,... random elements in Hilbert spaces The following theorem is an extension of the classical the Rademacher-Menshov strong law of large numbers to the blockwise PCND random elements in Hilbert spaces... negative dependence for random elements in Hilbert spaces 2.1 The strong law of large numbers and complete convergence In this section, we establish some strong laws of large numbers and complete
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