Summary of Mathematics Doctoral Thesis: Some limit theorems in noncommutative probability

Objective of the research: The research objective of the thesis is to establish some limit theorems regarding the forms of law of large numbers for sequences and arrays of measurable operators under different conditions.

MINISTRY OF EDUCATION AND TRAINING VINH UNIVERSITY -* - DO THE SON SOME LIMIT THEOREMS IN NONCOMMUTATIVE PROBABILITY Speciality: Theory of probability and mathematical statistics Code: 9460106 A SUMMARY OF MATHEMATICS DOCTORAL THESIS NGHE AN - 2020 Work is completed at Vinh University Supervisor: Prof Dr Nguyen Van Quang Dr Le Hong Son Reviewer 1: Reviewer 2: Reviewer 3: Thesis will be defended at university-level thesis evaluating council at Vinh University at , , Thesis can be found at: - Vietnam National Library - Nguyen Thuc Hao Library and Information Center - Vinh University PREFACE Rationale 1.1 The limit theorems have been interested in many mathematicians and have many applications in statistics, economics, medicine, and some other empirical sciences The limit theorems regarding the forms of law of large numbers are studied for many different objects For example, the law of large numbers for single-valued random variables, set-valued random variables, fuzzy set-valued random variables; the law of large numbers in game theory, in noncommutative probability In particular, the limit theorems in noncommutative probability are attracting the attention of many authors and have yielded certain results 1.2 The noncommutative integral theory was started studied in 1952-1953 by Segal Later, it was continued studied by Kunze (1958), Stinespring (1959), Nelson (1974), Yeadon (1979), etc On the basis of noncommutative integral theory, the theory of noncommutative probability has been studied by Batty (1979), Padmanabhan (1979), Luczak (1985), Jajte (1985) and is continuing to be of interest In noncommutative probability, there is no basic probability space, instead of studying random variables, we study operators on von Neumann algebra or measured operators Because multiplication of operators is not commutative and we cannot talk about the max, of operators, to study the problems of noncommutative probability theory, we are needed new tools, and techniques 1.3 The law of large numbers in noncommutative probability is studied in two main directions: bounded operators on von Neumann algebra with state and measured operators with tracial state The difficulty in the first direction is the limited nature of the state, while in the second, the unbounded property of the measurable operators give rise to many complex problems These characteristics contribute to the diversity of issues to be considered, studied of the limit theorems in noncommutative probability 1.4 As a result of many problems arising from quantum physics theory, problems of bounded operators on von Neumann algebra or measurable operators have been extensively studied from the seventies of the last century and continue to be studied up to now Therefore, the study of the limit theorems in noncommutative probability is not only theoretical sense but also practical sense With the above reasons, we have chosen the topic for the thesis as follows: “Some limit theorems in noncommutative probability” Objective of the research The research objective of the thesis is to establish some limit theorems regarding the forms of law of large numbers for sequences and arrays of measurable operators under different conditions Subject of the research The research subject of the thesis is the measurable operators and the law of large numbers for the measurable operators with the tracial state in noncommutative probability Scope of the research The thesis focuses on studying the law of large numbers for measurable operators in different types of convergence such as: convergence bilaterally almost uniformly, convergence in LP , convergence in measure; the study expanded the integrable concepts into noncommutative probability space Methodology of the research We use a combination of the fundamental methods of probability theory in proving the law of large numbers and the techniques of operator theory such as truncation method, subsequences method, spectral representation technique of the operators Contributions of the thesis The results of the thesis contribute to enriching the research direction of the limit theorems in noncommutative probability theory 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 In this thesis, we study limit theorems regarding the forms of law of large numbers for sequences and arrays of measurable operators For strong law of large numbers, we first establish some strong law for sequences of positive measurable operators Using these results, we prove some strong laws of large numbers for sequences of pairwise independent measurable operators that are identically distributed or non-identically distributed Next, we prove the equivalence conditions of uniform integrability for a sequence of measurable operators Based on these results, we introduce some integrable concepts for sequences of measurable operators in noncommutative probability Finally, the strong law of large numbers for sequences of pairwise independent measurable operators and strongly Ces`aro α-integrable is studied by us For weak law of large numbers, we first study the convergence in L1 for sequences of measurable operators, residually Ces`aro α-integrable and pairwise independent or m-dependent We then present the concepts: uniformly integrable in the Ces`aro sense, h-integrable with respect to the array of constant {ani } and h-integrable with exponent r of the array of measurable operators Finally, we establish some mean convergence theorems and weak law large numbers for arrays of measurable operators from the above concepts 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 regarding the forms of strong law of large numbers for sequences of measurable operators Chapter studies some limit theorems regarding the forms of weak law of large numbers for sequences and arrays of measurable operators 4 CHAPTER PRELIMINARIES In this chapter, we introduce some basic knowledge of noncommutative probability theory 1.1 The operators on a Hilbert space Definition 1.1.1 Let D be a subspace of H, the linear transformation T : D → H is called partially defined operator on H A densely defined operator on H with domain D(T ) is a partially defined operator on H whose domain D(T ) is dense in H A partially defined operator (or densely defined operator) may be bounded or unbounded A partially densely defined operator on H is closed if its graph is closed in H × H Theorem 1.1.2 If T is a self-adjoint, densely defined operator on H then there exist unique resolution E of the identity, on the Borel subsets of the real line, such that +∞ λdEx,y (λ) T (x), y = (x ∈ D(T ), y ∈ H) (1.1) −∞ Moreover, E is concentrated on σ(T ) ⊂ (−∞, +∞), in the sense that E σ(T ) = Formula (1.1) is called spectral representation of operator T and usually written as: +∞ T = +∞ λdE(λ) or T = −∞ λedλ (T ) −∞ Resolution E of the identity in theorem 1.1.2 is called spectral decomposition of operator T , and E(B) is spectral projection of operator T corresponding to a Borel subset B of the real line R, and write E(B) = eB (T ) 5 1.2 Von Neumann algebra Definition 1.2.1 A subalgebra A of L(H) is called von Neumann algebra if: i) A is self-adjoint, i.e, if T ∈ A then T ∗ ∈ A ; ii) A contains the identity operator 1; iii) A is weakly closed, i.e, if {Ti } ⊂ A is a net such that Ti → T in weak operator topology, then T ∈ A Definition 1.2.2 Let A ⊂ L(H) be a von Neumann algebra and τ : A → C be linear functional Denote A+ = {X ∈ A : X ≥ 0} Then i) τ is called positive if τ (X) ≥ 0, ∀X ∈ A+ ii) τ is called faithful if τ (X) = implies X = for all X ∈ A+ iii) τ is called state if τ positive and τ (1) = iv) State τ is called normal if, for all the net {Xi } ⊂ A+ , Xi ↑ X (in strong operator topology), then τ (Xi ) ↑ τ (X) v) State τ is called tracial state if τ (XY ) = τ (Y X), ∀X, Y ∈ A; τ (p ∨ q) τ (p) + τ (q), ∀p, q ∈ P rojA 1.3 Measurable operator Definition 1.3.1 Let X be a densely defined closed operator on H, and X have polar decomposition X = U |X| Then, operator X is called affiliated with von Neumann algebra A if U and spectral projections of |X| belong to A Denote A for the set of operator which affiliated to the von Neumann algebra A An element of A is called a measurable operator 6 Definition 1.3.2 Let A ⊂ L(H) be a von Neumann algebra and τ be a faithful normal tracial state on A Denote LP (A, τ ) (P ≥ 1) is a Banach space of all elements in A satisfying ||X||P = [τ (|X|P )] P < ∞ For notational consistency, A will be denoted by L0 (A, τ ) Then we have natural inclusions: A ≡ L∞ (A, τ ) ⊂ LQ (A, τ ) ⊂ LP (A, τ ) ⊂ ⊂ L0 (A, τ ) = A, for ≤ P ≤ Q < ∞ 1.4 Some kinds of convergence and independence In this section, let A ⊂ L(H) be a von Neumann algebra, τ be a faithful normal tracial state on A, and L0 (A, τ ) be an algebra of measurable operator Definition 1.4.1 The sequence {Xn , n ≥ 1}) ⊂ L0 (A, τ ) is called converges in measure to τ X ∈ L0 (A, τ ), denoted by Xn − → X, if any ε > 0, τ e(ε,∞) (|Xn − X|) −→ as n −→ ∞ Definition 1.4.2 The sequence {Xn , n ≥ 1} ⊂ L0 (A, τ ) is called converges in LP to X ∈ LP L0 (A, τ ), denoted by Xn −−→ X, if τ |Xn − X|P −→ as n −→ ∞ Definition 1.4.3 The sequence {Xn , n ≥ 1} ⊂ L0 (A, τ ) is called converges almost uniformly a.u to X ∈ L0 (A, τ ), denoted by Xn −−→ X, if for each ε > 0, there exists a projection p ∈ A such that τ (p⊥ ) < ε, (Xn − X)p ∈ A, and lim (Xn − X)p n→∞ ∞ = Definition 1.4.4 The sequence {Xn , n ≥ 1} ⊂ L0 (A, τ ) is called converges bilaterally almost b.a.u uniformly to X ∈ L0 (A, τ ), denoted by Xn −−−→ X, if for each ε > 0, there exists a projection p ∈ A such that τ (p⊥ ) < ε, p(Xn − X)p ∈ A, and lim p(Xn − X)p n→∞ The conclusions of Chapter In this chapter, we obtain some main results: ∞ = 7 - A brief some of the basic concepts and properties of operators on a Hilbert space; - Prove some properties of measurable operators; - System some kinds of convergence in noncommutative probability and the relationship between them; - Present some independent concepts of sequences and arrays of measurable operators 8 CHAPTER SOME LIMIT THEOREMS REGARDING THE FORMS OF STRONG LAW OF LARGE NUMBERS FOR SEQUENCES OF MEASURABLE OPERATORS In this chapter, we establish some limit theorems regarding the forms of strong law of large numbers (with convergence bilaterally almost uniformly) for sequences of measured operators 2.1 The strong law of large numbers for sequences of positive measurable operators In this section, we study some strong law of large numbers for sequences of positive measurable operators The following theorem is an extension of Theorem from Chandra et al (1992) to noncommutative probability Theorem 2.1.1 Let {Xn , n ≥ 1} ⊂ L0 (A, τ ) be a sequence of positive operators satisfying the following conditions: τ (Sn ) < ∞, (i) sup n≥1 f (n) (ii) there exists a double sequence {ρij } of nonnegative reals such that τ |Sn − τ (Sn )|2 ≤ n (3.1) n ρij for all n ≥ 1, (3.2) < ∞, i ∨ j = max(i, j) (3.3) i=1 j=1 ∞ ∞ ρij i=1 j=1 (f (i ∨ j))2 (iii) Then Sn − τ (Sn ) b.a.u −−−→ as n → ∞ f (n) Theorem 2.1.2 Let {Xn , n ≥ 1} ⊂ L0 (A, τ ) be a sequence of positive operators such that there is a sequence {Bn } of Borel subsets of R satisfying the following conditions: ∞ τ (eBnc (Xn )) < ∞ , (i) n=1 (3.7) n (ii) k=1 τ Xk eBkc (Xk ) = o (f (n)) , (3.8) τ (Sn ) 0, such that for any projection p in A, if τ (p) < δ then sup τ (|Xn |p) < , (2.2) n≥1 and, in addition sup τ (|Xn |) < ∞ n≥1 (2.3) 12 (iv) lim sup τ |Xn |e(a,∞) (|Xn |) = a→∞ n≥1 Definition 2.3.3 A sequence {Xn , n ≥ 1} of measurable operators is said to be Ces` aro Uniformly Integrable (CUI, in short) if lim sup c→∞ n≥1 n n τ |Xk |e(c,∞) (|Xk |) = (2.4) k=1 Relying on Theorem 2.3.2, we can easily show that the UI of {Xn , n ≥ 1} implies the CUI of {Xn , n ≥ 1} The following proposition is the noncommutative analogue of the clasiccal criterion for Ces` aro uniform integrability of random variables due to Chandra (1992) Proposition 2.3.4 (noncommutative criterion for Ces` aro uniform integrability) Let {Xn , n ≥ 1} be a sequence of measurable operators The following properties are equivalent: (i) {Xn , n ≥ 1} is CUI (ii) there exists a convex* function φ ∈ Φ, such that sup n≥1 n n τ [φ(|Xk |)] = M < ∞ (2.5) k=1 * This property is not used for (ii) ⇒ (i) It follows from Proposition 2.3.4 that if {Xn , n ≥ 1} is SCUI then it is CUI Definition 2.3.5 Let α ∈ (0, ∞) A sequence {Xn , n ≥ 1} of measurable operators is called (i) Ces` aro α-Integrable (CI(α), in short) if the following two conditions hold: n n τ |Xi | < ∞ τ |Xi |e(iα ,∞) (|Xi |) = sup and lim n→∞ n i=1 n≥1 n i=1 (ii) Strongly Ces` aro α-Integrable (SCI(α), in short) if the following two conditions hold: n ∞ 1 sup τ |Xi | < ∞ and τ |Xn |e(nα ,∞) (|Xn |) < ∞ n≥1 n i=1 n=1 n Obviously, if < α < β then CI(α) implies CI(β) and SCI(α) implies SCI(β) Moreover, the Kronecker lemma shows that SCI(α) implies CI(α), for all α > We now prove that for all α > 0, CUI =⇒ CI(α) and SCUI =⇒ SCI(α) Lemma 2.3.6 Suppose that {Xn , n ≥ 1} is a sequence of measurable operators and let α be a positive real 13 (i) If {Xn , n ≥ 1} is CUI, then it is CI(α) (ii) If {Xn , n ≥ 1} is SCUI, then it is SCI(α) Definition 2.3.7 Let α ∈ (0, ∞) A sequence {Xn , n ≥ 1} of measurable operators is called (i) Residually Ces` aro α-Integrable (RCI(α), in short) if the following two conditions hold: n n τ |Xi | < ∞ and lim τ |Xi | − iα e(iα ,∞) (|Xi |) = sup n→∞ n n n≥1 i=1 i=1 (ii) Strongly Residually Ces` aro α-Integrable (SRCI(α), in short) if the following two conditions hold: n sup τ |Xi | n≥1 n i=1 ∞ 0, SRCI(α) =⇒ RCI(α) by the Kronecker’s lemma It follows from the definitions that if < α < β then RCI(α) =⇒ RCI(β) and SRCI(α) =⇒ SRCI(β) Furthermore, CI(α) =⇒ RCI(α) and SCI(α) =⇒ SRCI(α), for all α > Theorem 2.3.8 Let {Xn , n ≥ 1} ⊂ L0 (A, τ ) be a sequence of pairwise independent, selfadjoint operators If the sequence {Xn , n ≥ 1} satisfies the condition SCI(α) for some α ∈ (0, ), then Sn − τ (Sn ) b.a.u −−−→ as n → ∞ n Lemma 2.3.6 indicates that for any α > 0, SCI(α) is weaker than SCUI Thus, we get the following corollary that is a noncommutative versions of Theorem in Chandra (1992) Corollary 2.3.9 Let {Xn , n ≥ 1} ⊂ L0 (A, τ ) be a sequence of pairwise independent, selfadjoint operators If the sequence {Xn , n ≥ 1} satisfies the condition SCUI , then Sn − τ (Sn ) b.a.u −−−→ as n → ∞ n The conclusions of Chapter In this chapter, we obtain some main results: - Establish some strong law of large nubers for sequences of positive measurable operators; - Prove some strong laws of large numbers for pairwise independent non-identically distributed measurable operators and for pairwise independent identically distributed measurable operators; 14 - Establish some equivalent conditions of uniformly integrability for a sequences of measurable operators; - Construct some integrable concepts in noncommutative probability and the relationship between them; - Prove some strong laws for sequences of pairwise independent measurable operators and satisfies condition strongly Ces`aro α-integrable 15 CHAPTER SOME LIMIT THEOREMS REGARDING THE FORMS OF WEEK LAWS OF LARGE NUMBERS FOR SEQUENCES AND ARRAYS OF MEASURABLE OPERATORS In this chapter, we study some limit theorems regarding the forms of week laws of large numbers for sequences and arrays of measurable operators 3.1 The weak laws of large numbers for sequences of measurable operators In this section, we will establish some convergence theorems in L1 We know that the convergence in L1 implies the convergence in measure Therefore, from the theorems in this section, we can obtain the results regarding the forms of the weak law of large numbers for sequences of measurable operators Theorem 3.1.1 Let {Xn , n ≥ 1} ⊂ L0 (A, τ ) be a sequence of pairwise independent, selfadjoint operators If the sequence {Xn , n ≥ 1} satisfies the condition RCI(α) for some α ∈ (0, 1), then Sn − τ (Sn ) → in L1 as n → ∞ n By applying Theorem 3.1.1 we obtain the following corollary immediately: Corollary 3.1.2 [Theorem 4.1(b), Lindsay and Pata (1997)] Let {Xn , n ≥ 1} ⊂ L0 (A, τ ) be a sequence of pairwise independent, self-adjoint operators For each n ≥ 1, put Sn = n Xi e[0,n] (|Xi |) If the sequence {Xn , n ≥ 1} satisfies: i=1 K(a) := sup a Xi e(a,∞) (|Xi |) i → as a → ∞, then Sn − τ (Sn ) → in L1 as n → ∞ n 16 As in clasiccal situation, we say that a sequence {Xn , n ≥ 1} of measurable operators are pairwise m-dependent, if Xi and Xj are independent whenever |i − j| > m Theorem 3.1.3 Let {Xn , n ≥ 1} ⊂ L0 (A, τ ) be a sequence of pairwise m-dependent, selfadjoint operators If the sequence {Xn , n ≥ 1} satisfies the condition RCI(α) for some α ∈ (0, 1), then Sn − τ (Sn ) → in L1 as n → ∞ n 3.2 The weak laws of large numbers for arrays of measurable operators In this section, we introduce the notions of uniform integrability Then, we establish some mean convergence theorems and weak laws of large numbers for arrays of measurable operators under some conditions related to these notions We now provide some lemmas which will be helpful in obtaining main results The proof of the following Lemma 3.2.1 is the same as that of Lemma 2.1 in Wu and Guan (2011) and is omitted Lemma 3.2.1 Let {Xni , un ≤ i ≤ , n ≥ 1} be an array of measurable operators and r > Let moreover {h(n), n ≥ 1} be an increasing sequence of positive constants with h(n) ↑ ∞ as n → ∞ Let {ani , un ≤ i ≤ , n ≥ 1} be an array of constans satisfying h(n) supun ≤i≤vn |ani |r → as n → ∞ Suppose the following conditions hold: |ani |r τ (|Xni |r ) < ∞, (i) sup (3.1) n≥1 i=un |ani |r sup yτ e(y1/r ,∞) (|Xni |) = (ii) lim n→∞ i=u Then for all β > r (3.2) y≥h(n) n |ani |β τ |Xni |β e[0,kn1/r ] (|Xni |) = 0, lim n→∞ i=un where kn = 1/ supun ≤i≤vn |ani |r −1/r Taking ani = kn lemma for un ≤ i ≤ and n ≥ in Lemma 3.2.1, we can get the following 17 Lemma 3.2.2 Let {Xni , un ≤ i ≤ , n ≥ 1} be an array of measurable operators and r > Let moreover {h(n), n ≥ 1} be an increasing sequence of positive constants with h(n) ↑ ∞ as h(n) → 0, and the following conditions hold: n → ∞ Suppose that kn → ∞, kn (i) sup τ (|Xni |r ) < ∞, (3.3) n≥1 kn i=un n→∞ kn sup yτ e(y1/r ,∞) (|Xni |) = (ii) lim (3.4) i=un y≥h(n) Then for all β > r lim n→∞ −β/r kn τ |Xni |β e[0,kn1/r ] (|Xni |) = i=un Now, we present the main results of this section Under some appropriate conditions, some mean convergence theorems and weak laws of large numbers for arrays of measurable operators will be established Theorem 3.2.3 Let {Xni , un ≤ i ≤ , n ≥ 1} be an array of rowwise pairwise independent h(n) → If the measurable operators and ≤ r < Suppose that kn → ∞, h(n) ↑ ∞, and kn conditions (3.3) and (3.4) hold, then τ 1/r kn i=un → n → ∞ Xni − τ (Xni ) − Theorem 3.2.4 Let {Xni , un ≤ i ≤ , n ≥ 1} be an array of rowwise pairwise independent measurable operators and let {ani , un ≤ i ≤ , n ≥ 1} be an array of non-negative constants, h(n) kn = 1/ sup ani Suppose that h(n) ↑ ∞, and → If (3.1), (3.2) hold with r = 1, kn un ≤i≤vn and the following condition: ani τ |Xni |e(kn ,∞) (|Xni |) = 0, lim n→∞ i=u is satisfied, then (4.1) n L1 ani Xni − τ (Xni ) −→ as n → ∞ i=un The following theorem shows that under some conditions stronger than those of Theorem 3.2.3, we can obtain r-mean convergence for the array of rowwise pairwise independent measurable operators Theorem 3.2.5 Let {Xni , un ≤ i ≤ , n ≥ 1} be an array of rowwise pairwise independent measurable operators and ≤ r < 2, and let {ani , un ≤ i ≤ , n ≥ 1} be an array of nonnegative constants kn = 1/ supun ≤i≤vn arni , h(n) ↑ ∞, and h(n) kn → Suppose (3.1), (3.2), and 18 i=un arni τ |Xni |2 e(kn1/r ,∞) (|Xni |) = O(log δ kn ), (4.2) holds for some δ > Then Lr ani Xni − τ (Xni ) −→ as n → ∞ i=un The following theorem is an extension of Theorem 3.1 in Ankirchner et al (2017) to noncommutative probability Theorem 3.2.6 Let {Xni , ≤ i ≤ kn , n ≥ 1} be an array of rowwise pairwise independent positive measurable operators Suppose that kn (i) lim n→∞ τ e[kn ,∞) (Xni ) = 0, (4.3) i=1 and kn (ii) lim n→∞ z τ e[kn z,∞) (Xni ) i=1 τ e[0,kn u) (Xni ) dudz = (4.4) Then kn kn τ Xni − τ Xni e[0,kn ) (Xni ) − → as n → ∞ i=1 The following Corollary is stronger than Theorem 3.1 in Luczak (1985) because the condition “successively independent” is replaced by the weaker one ”pairwise independent” Corollary 3.2.7 Let {Xn , n ≥ 1} be a pairwise independent sequence of self-adjoint, identically distributed elements from L0 (A, τ ) If lim nτ e[n,∞) (|X1 |) = 0, n→∞ then n (4.6) n τ → as n → ∞ Xi − τ X1 e[0,n) (|X1 |) − i=1 The conclusions of Chapter In this chapter, we obtain some main results: - Establish a weak law of large numbers for sequences of self-adjoint measurable operators and residually Ces`aro α-integrable in case of pairwise independent or m-dependent; - Construct some integrable concepts for arrays of measurable operators; - Prove some weak laws of large numbers for arrays of rowwise pairwise independent measurable operators 19 GENERAL CONCLUSIONS AND SUGGESTIONS General conclusions In this thesis, we obtain some main results: - Establish some equivalence conditions of uniform integrability for a sequence of measurable operators; - Construct some integrable concepts for sequences of measurable operators and the relationship between them in noncommutative probability; - Prove some integrable criterion in noncommutative probability; - Establish some strong laws of large numbers for sequences of measurable operators that are positive or pairwise independent of one; - Construct some integrable concepts for arrays of measurable operators and establish some weak laws of large numbers for sequences, arrays of measurable operators Recommendations In the near future we will study the following issues: - Establish strong laws of large numbers and complete convergence for arrays of rowwise successively measurable operators; - Expand some Baum-Katz type strong laws of large numbers to noncommutative probability; - Study on free random variables 20 LIST OF THE AUTHOR’S ARTICLES RELATED TO THE THESIS Nguyen Van Quang, Do The Son, Le Hong Son (2017), The strong laws of large numbers for positive measurable operators and applications, Statistics and Probability Letters, 124, 110-120 Nguyen Van Quang, Do The Son, Le Hong Son (2018), Some kinds of uniform integrability and laws of large numbers in noncommutative probability, Journal of Theoretical Probability, 31, 1212-1234 Nguyen Van Quang, Do The Son, Tien-Chung Hu, Nguyen Van Huan (2019), Mean convergence theorems and weak laws of large numbers for arrays of measurable operators under some conditions of uniform integrability, Lobachevskii Journal of Mathematics, 40, 1218-1229 Results of the thesis have been reported in: - The 9th Vietnam Mathematics Congress (University of Information and Communications, Nha Trang, August 14-18, 2018); - 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)  ... follows: ? ?Some limit theorems in noncommutative probability? ?? Objective of the research The research objective of the thesis is to establish some limit theorems regarding the forms of law of large... variables; the law of large numbers in game theory, in noncommutative probability In particular, the limit theorems in noncommutative probability are attracting the attention of many authors and... SOME LIMIT THEOREMS REGARDING THE FORMS OF STRONG LAW OF LARGE NUMBERS FOR SEQUENCES OF MEASURABLE OPERATORS In this chapter, we establish some limit theorems regarding the forms of strong law of