Một số dạng luật số lớn cho dãy và mảng các biến ngẫu nhiên trong không gian tổ hợp lồi (tt)

1 PREFACE Rationale In recent decades, some results on limit theorems of law of large numbers for a collection of random variables in metric space have been researched and established by several authors In 1992, Herer introduced a concept of mathematical expectation of random variables in a separable and complete metric space (X, d) of negative curvature Then, Herer derived strong law of large numbers for sequences of independent and identically distributed integrable random variables In 1997, using Herer’s definition of the mathematical expectation of random variables in X and the approximation method by a sequence of discrete random variables, Fitte derived an ergodic theorem and strong law of large numbers for sequences of integrable random variables Some limit theorems of martingale in a metric space established by Herer (in 1992, 1997) and Sturm (in 2002) In 2006, Ter´an and Molchanov introduced the concept of convex combination space, which is a metric space endowed with a convex combination operation Then, Ter´an and Molchanov constructed the concept of mathematical expectation of random variables with values in a convex combination space and derived strong law of large numbers for sequences of pairwise independent and identically distributed random variables Hence, the researching on limit theorems of the law of large numbers for random variables in metric space is an up-to-date tendency of the probability theory The limit theorems on the law of large numbers and complete convergence also research for double arrays and triangular arrays of random variables You can find the basic results of this area in the monographic book of Klesov (in 2014) Note that, when extending the theorems for sequences of random variables to arrays, then the results and methods used for the sequences are not always applicable to the arrays Hence, the results on the researching of limit theorems for double arrays and triangular arrays of random variables in metric space are interesting and more meanings When researching the strong law of large numbers, people often consider the independence of random variables A research direction on limit theorems of the law of large numbers is to substitute independent conditions with weaker conditions such as pairwise independent, blockwise m-dependent, blockwise and pairwise m-dependent This is a research direction that deserves attention Researching the limit theorems of fuzzy random variables is very important in theory and practice In practice, the theory of fuzzy random variables has been researched and applied to such areas: information technology, image processing, control engineering and some other areas In theory, many problems in theory of fuzzy random variables relate to classical probability theory Some the limit theorems in classical probability theory were extended to fuzzy random variables Especially, the law of large numbers for fuzzy random variables has been studied by many researchers For example, Colubi (in 1999) derived the strong law of large numbers for independent and identically distributed fuzzy random variables in Rd Proske and Puri (in 2002) proved the strong law of large numbers for independent and identically distributed fuzzy random variables in Banach space Inoue (in 1991) obtained the law of large numbers for sums of independent tight fuzzy random variables, which extends the result of Taylor and Inoue (in 1985) for set-valued random variables Recently, Kim (in 2013) established weak law of large numbers for the weighted sum of fuzzy random variables taking values in a real separable Banach space Hence, researching the limit theorems of the law of large numbers for fuzzy random variables in metric space is a research direction that has many meanings and values With the above reasons, we have chosen the topic for the thesis that is: “Some types of laws of large numbers for sequences and arrays of random variables in convex combination space” Objective of the research The objective of the thesis is to establish the strong law of large numbers for sequences of random variables and arrays of fuzzy random variables in convex combination space, to establish complete convergence and strong law of large numbers for triangular arrays of random variables and triangular arrays of fuzzy random variables in convex combination space, to establish mean convergence and weak law of large numbers for sequences of fuzzy random variables in convex combination space under various assumptions 3 Subject of the research - The strong law of large numbers for sequences of random variables, double arrays of fuzzy random variables - The complete convergence and strong law of large numbers for triangular arrays of random variables, triangular arrays of fuzzy random variables - The mean convergence and weak law of large numbers for sequences of fuzzy random variables Scope of the research Some types of convergence of sequences and arrays of random variables in convex combination space The types of convergence are considered: almost sure convergence, complete convergence, mean convergence and convergence in probability Methodology of the research We use the methods of probability and characteristics of analysis as: approximation method, truncation method, properties of a compact set, etc Contribution of the thesis The results of thesis contribute more abundant for the researching directions of general limit theorems and limit theorems for random variables in metric space The thesis is contribution of material for the students, the master students, the doctoral students belonging to the speciality Theory of Probability and Mathematical Statistics Organization of the research 7.1 Overview of the research In this thesis, using the definition of convex combination space of Ter´an and Molchanov (in 2006), we establish limit theorems on the strong law of large numbers, complete convergence, mean convergence and weak law of large numbers for sequences and arrays of random variables and fuzzy random variables in convex combination space First, we present some concepts and basic properties of convex combination space Next, we present concepts of compactly uniformly integrable and compactly uniformly r-th integrable in Ces`aro sense for a collection of random variables in convex combination space, which are naturally extended from the corresponding concepts in Banach space to convex combination space We establish the strong law of large numbers for sequences of random variables in convex combination space satisfying the condition: blockwise and pairwise m-dependent and compactly uniformly integrable in Ces`aro sense, or blockwise m-dependent and identically distributed We also establish the complete convergence and the strong law of large numbers for triangular arrays of rowwise independent and compactly uniformly integrable random variables in convex combination space For triangular arrays of fuzzy random variables in convex combination space, we establish the complete convergence and the strong law of large numbers for rowwise independent and (α, α+ )-levelwise compactly uniformly integrable fuzzy random variables For double arrays of fuzzy random variables in convex combination space, we establish the strong law of large numbers for fuzzy random variables satisfying the condition: independent and (α, α+ )-levelwise compactly uniformly integrable, or pairwise independent and identically distributed Finally, we establish necessary and sufficient conditions on mean convergence and weak law of large numbers for sequences of fuzzy random variables The convergence of fuzzy random variables in this thesis is considered by metric d∞ 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 basic knowledge of convex combination space and random variable in convex combination space This chapter is organized as follows: Section 1.1 presents concept of convex combination space, some examples of convex combination space, basic properties of convex combination space Section 1.2 presents random variable in convex combination space, defines the mathematical expectation and integrability of a random variable in convex combination space, types of convergence: almost sure convergence, complete convergence, mean convergence and convergence in probability for sequences and arrays of random variables in convex combination space, concept of compactly uniformly integrable and compactly uniformly r-th integrable in Ces`aro sense for a collection of random variables in convex combination space Section 1.3 presents concept and basic properties of fuzzy random variable in convex combination space The knowledge of Chapter is used to establish the main results of the next chapters Chapter presents some limit theorems on the law of large numbers for sequences and triangular arrays of random variables in convex combination space Section 2.1 presents the strong law of large numbers for sequences of blockwise and pairwise m-dependent random variables in convex combination Section 2.2 presents the complete convergence and strong law of large numbers for triangular arrays of random variables in convex combination space Chapter is used to research some limit theorems on the law of large numbers for sequences, triangular arrays and double arrays of fuzzy random variables in convex combination space Section 3.1 presents concepts of (α, α+ )-levelwise compactly uniformly integrable and (α, α+ )-levelwise compactly uniformly r-th integrable in Ces`aro sense for a collection of fuzzy random variables in convex combination space Section 3.2 presents the complete convergence and the strong law of large numbers for triangular arrays of fuzzy random variables in convex combination space Section 3.3 presents the strong law of large numbers for double arrays of fuzzy random variables in convex combination space Section 3.4 presents the mean convergence and weak law of large numbers for sequences of fuzzy random variables in convex combination space 6 CHAPTER THE PREPARATION KNOWLEDGE In this chapter, we present some concepts and basic properties of convex combination pace, some examples of convex combination space, concepts and properties of a random variable and a fuzzy random variable in convex combination space We present the concept of compactly uniformly integrable and compactly uniformly r-th integrable in Ces`aro sense for a collection of random variables in convex combination space 1.1 Convex combination space In this thesis, if not added assume, we suppose that (Ω, A, P ) is a complete probability space, (X, d) is a separable and complete metric space, BX is the Borel σ-algebra on X Let c(X) be the family of all nonempty compact subsets of X We denoted by N (resp N0 ) (resp R) the set of all positive integers (resp nonnegative integers) (resp real numbers) For two real numbers m and n, we denote max{m, n} (resp min{m, n}) by m ∨ n (resp m ∧ n) For each a ∈ R+ , the logarithm to the base of a ∨ will be denoted by log+ a On the metric space (X, d), we define a convex combination operation: for all n 2, numbers λ1 , , λn > that satisfy n i=1 λi = and all u1 , , un ∈ X, this operation produces an element of X, which is denoted by [λi , ui ]ni=1 or [λ1 , u1 ; ; λn , un ] Assume that [1, u] = u for every u ∈ X and that the following axioms are satisfied • Axiom 1: (Commutativity) [λi , ui ]ni=1 = [λσ(i) , uσ(i) ]ni=1 for every permutation σ of {1, , n}; • Axiom 2: (Associativity) λ n+j [λi , ui ]n+2 i=1 = [λ1 , u1 ; ; λn , un ; λn+1 + λn+2 , [ λn+1 +λn+2 , un+j ]j=1 ]; • Axiom 3: (Continuity) If u, v ∈ X and λ(k) → λ ∈ (0; 1) as k → ∞, then [λ(k) , u; − λ(k) , v] → [λ, u; − λ, v]; • Axiom 4: (Negative curvature) If u1 , u2 , v1 , v2 ∈ X and λ ∈ (0; 1), then d([λ, u1 ; − λ, u2 ], [λ, v1 ; − λ, v2 ]) λd(u1 , v1 ) + (1 − λ)d(u2 , v2 ); • Axiom 5: (Convexification) For each u ∈ X, there exists lim [n−1 , u]ni=1 , n→∞ which will be denoted by KX u (or Ku so no confusion can arise) and K is called the convexification operator 1.1.1 Definition The metric space (X, d) endowed with a convex combination operation is referred to as the convex combination space Note that, in the general case, Ku and u are not identical An element u ∈ X is called convexely decomposable element if for all n n i=1 λi and λ1 , , λn > with = 1, then u = [λi , u]ni=1 A set A ⊂ X is called convex if [λi , ui ]ni=1 ∈ A for all ui ∈ A and {λi : i n, n n i=1 λi 1} ⊂ (0; 1) satisfying = For A ⊂ X, we denote as coA the convex hull of A, which is the smallest convex subset that contains A, and coA is the closed convex hull of A 1.1.11 Theorem If (X, d) is a convex combination space, then the space c(X) with the convex combination [λi , Ai ]ni=1 = {[λi , ui ]ni=1 : ui ∈ Ai , for all i} and the Hausdorff metric dH dH (A, B) = max sup inf d(a, b), sup inf d(b, a) a∈A b∈B b∈B a∈A is also a convex combination space, where the convexification operator Kc(X) is given by Kc(X) A = coKX A = co{KX u : u ∈ A} Since (X, d) is a separable and complete metric space, (c(X), dH ) is also a separable and complete metric space 8 1.2 Random variable in convex combination space From now on, we assume that (X, d) is a convex combination space 1.2.1 Definition A mapping X : Ω → X is called A-measurable if for all B ∈ BX , then X −1 (B) ∈ A The mapping A-measurable X is also called an X-valued random variable When an X-valued random variable X takes finite values, it is called a simple random variable For each X-valued random variable X, we denote σ(X) = {X −1 (B) : B ∈ BX } Then, σ(X) is the smallest sub-σ-algebra of A with respect to which X is measurable The distribution of X-valued random variable X is a probability measure PX on BX defined by PX (B) = P X −1 (B) , B ∈ BX 1.2.2 Definition The collection of X-valued random variables {Xi : i ∈ I} is said to be independent (resp pairwise independent) if the collection of σalgebras {σ(Xi ) : i ∈ I} is independent (resp pairwise independent), and is said to be identically distributed if all PXi , i ∈ I, are identical 1.2.3 Definition (a) The sequence of X-valued random variables {Xn : n 1} is said to be almost sure convergence to the X-valued random variable X as n → ∞, if there exists A ∈ A such that P (A) = and for all ω ∈ Ω \ A, then lim d(Xn , X)(ω) = n→∞ a.s We denote Xn → X a.s (or Xn −→ X) as n → ∞ (b) The double array of X-valued random variables {Xmn : m 1, n 1} is said to be almost sure convergence to the X-valued random variable X-valued random variable X as m ∨ n → ∞, if there exists A ∈ A such that P (A) = and for all ω ∈ Ω \ A, then lim m∨n→∞ d(Xmn , X)(ω) = a.s We denote Xmn → X a.s (or Xmn −→ X) as m ∨ n → ∞ (c) The sequence of X-valued random variables {Xn : n 1} is said to be complete convergence to the X-valued random variable X as n → ∞, if for all ε > 0, then ∞ P d(Xn , X) > ε < ∞ n=1 c We denote Xn −→ X as n → ∞ (d) The sequence of X-valued random variables {Xn : n 1} is said to be convergence in p-th mean, p > 0, to the X-valued random variable X as n → ∞, if lim Edp (Xn , X) = n→∞ Lp We denote Xn −→ X as n → ∞ (e) The sequence of X-valued random variables {Xn : n 1} is said to be convergence in probability to the X-valued random variable X as n → ∞, if for all ε > 0, then lim P d(Xn , X) > ε = n→∞ P We denote Xn −→ X as n → ∞ Let X be an X-valued random variable that takes a distinct value xi for each non-null set Ωi with i = 1, , n The expectation of X is defined by EX = [P (Ωi ), Kxi ]ni=1 (1.2.1) Note that, if X, Y are simple random variables then d(EX, EY ) Ed(X, Y ) We fix u0 ∈ K(X) (by Axiom 5, K(X) = ∅) and u0 will be considered as the special element of X Since the metric space X is separable, there exists a countable dense subset {un : n 1} of X For each k 1, we define the mapping ϕk : X → X such that ϕk (x) = umk (x) , where mk (x) = i ∈ {0, , k} : d(ui , x) = d(uj , x) j k Hence, we obtain d(u0 , ϕk (x)) 2d(u0 , x) 1.2.5 Definition The X-valued random variable X is said to be integrable if d(u0 , X) is the integrable real-valued random variable The space of all integrable X-valued random variables will be denoted by L1X Then, for X ∈ L1X , the expectation of X is defined by EX := lim Eϕk (X) k→∞ 10 By the approximation method, we also prove that if X, Y ∈ L1X , then d(EX, EY ) Ed(X, Y ) The Borel σ-algebra on c(X) is generated by the collection of sets KV := {A ∈ c(X) : A ∩ V = ∅}, where V is an open subset in X, and denoted by Bc(X) 1.2.7 Definition A mapping X : Ω → c(X) is called c(X)-valued random variable if for all B ∈ Bc(X) , then X −1 (B) ∈ A Note that (c(X), dH ) ia a separable and complete convex combination space, therefore the concepts and properties of the expectation of integrable c(X)valued random variables are similar as integrable X-valued random variables We denote the expectation of an integrable c(X)-valued random variable X by Ec(X) X 1.2.8 Definition A collection {Xi : i ∈ I} of X-valued (resp c(X)-valued) random variables is said to be compactly uniformly integrable (CUI, for short) if for every ε > 0, there exists a compact subset Kε of X (resp c(X)) such that sup E d(u0 , Xi )I{Xi ∈ / Kε } ε i∈I resp sup E dH {u0 }, Xi I{Xi ∈ / Kε } ε i∈I 1.2.9 Definition A sequence {Xn : n 1} of X-valued (resp c(X)-valued) random variables is said to be compactly uniformly r-th integrable in Ces`aro sense, r > (Ces`aro r-th CUI, for short) if for every ε > 0, there exists a compact subset Kε of X (resp c(X)) such that n −1 E dr (u0 , Xi )I{Xi ∈ / Kε } sup n n ε i=1 n −1 E drH {u0 }, Xi I{Xi ∈ / Kε } resp sup n n ε i=1 Especially, when r = 1, we also call compactly uniformly integrable in Ces`aro sense (Ces`aro CUI, for short) 11 A collection {Xi : i ∈ I} of real-valued random variables is said to be uniformly bounded by a real-valued random variable X, if for all i ∈ I and t 0, then P (|Xi | t) P (|X| t) A collection {Xi : i ∈ I} of real-valued random variables is said to be stochastically dominated by a real-valued random variable X, if there exists a constant C (0 < C < ∞) such that for all i ∈ I and t P (|Xn | t) CP (|X| 0, then t) 1.3 Fuzzy random variable in convex combination space A mapping v : X → [0; 1] is said to be fuzzy set on X We denote by F(X) the space of all fuzzy set v that satisfy: v is upper semicontinuous, sup v = and supp v = cl{x ∈ X : v(x) > 0} is compact in X For v ∈ F(X), its α-level set Lα v is defined Lα v = {x ∈ X : v(x) α}, with α ∈ (0; 1] Moreover, we also define L+ α v = cl{x ∈ X : v(x) > α}, with α ∈ [0; 1) We have Lα v and L+ α v are compact in X Furthermore, we can check that Lβ v, for all α ∈ (0; 1] Lα v = (1.3.1) βα Especially, L+ v = supp v 1.3.1 Theorem F(X) with the convex combination operator given by Lα ([λi , vi ]ni=1 ) = [λi , Lα vi ]ni=1 , α ∈ (0; 1], vi ∈ F(X), and the metric d∞ (v1 , v2 ) = sup dH (Lα v1 , Lα v2 ) α∈(0;1] (1.3.2) 12 is a convex combination space, where the convexification operator KF(X) is given by Lα (KF(X) v) = Kc(X) Lα v = coKX (Lα v), α ∈ (0; 1] 1.3.2 Theorem F(X) with the convex combination operator is described in Theorem 1.3.1 and the metric dp (v1 , v2 ) = dpH (Lα v1 , Lα v2 )dα 1/p , p is a convex combination space, where the convexification operator KF(X) is given by Lα (KF(X) v) = Kc(X) Lα v, α ∈ (0; 1] For v ∈ F(X), we denote v ∞ = d∞ (v, I{u0 } ) = supα>0 Lα v {u0 } , where I{u0 } is the upper semicontinuous function that takes values at u0 and for all x = u0 1.3.3 Definition A mapping X : Ω → F(X) is said to be a fuzzy random variable in convex combination space X if Lα X is a c(X)-valued random variable for all α ∈ (0; 1] From the above definition and (1.3.2) we have if X is a fuzzy random variable in convex combination space X, then L+ α X is a c(X)-valued random variable, for all α ∈ [0; 1) A fuzzy random variable in convex combination space X is called integrably bounded if L+ 0X {u0 } ∈ L1R , then we denote X ∈ L1 (F(X)) 1.3.5 Definition The expectation of X ∈ L1 (F(X)), denoted by EF(X) X, is a fuzzy set on X such that for each α ∈ (0; 1], then Lα (EF(X) X) = Ec(X) (Lα X) 1.3.7 Definition A collection {Xi : i ∈ I} of fuzzy random variables in convex combination space X is said to be independent (resp pairwise independent) if {Lα Xi : i ∈ I} is a collection of independent (resp pairwise independent) c(X)-valued random variable, for each α ∈ (0; 1] The conclusions of Chapter In this chapter, we present concepts and results about convex combination space, random variable in convex combination space, fuzzy random variable in 13 convex combination space, the concepts of compactly uniformly integrable and compactly uniformly r-th integrable in Ces`aro sense for a collection of random variables in convex combination space 14 CHAPTER SOME TYPES OF STRONG LAWS OF LARGE NUMBERS FOR SEQUENCES AND TRIANGULAR ARRAYS OF RANDOM VARIABLES IN CONVEX COMBINATION SPACE In this chapter, first we introduce some concepts relatively to sequences of blockwise and pairwise m-dependent random variables in convex combination space Next we establish some results on strong laws of large numbers for sequences of blockwise and pairwise m-dependent random variables in convex combination space Finally we present a result on complete convergence and the strong laws of large numbers for triangular arrays of random variables in convex combination space 2.1 Strong laws of large numbers for sequences of blockwise and pairwise m-dependent random variables in convex combination space Let m be a fixed nonnegative integer A finite collection {Xi : of X-valued random variables is said to be m-dependent if either n i n} m + 1; or n > m + and the random variables {X1 , , Xi } are independent of the random variables {Xj , , Xn } whenever j − i > m A finite collection {Xi : i be pairwise m-dependent if either n n} of X-valued random variables is said to m + 1; or n > m + and Xi and Xj are independent whenever j − i > m A sequence {Xn : n 1} of X-valued random variables is said to be pairwise m-dependent if Xi and Xj are independent whenever j − i > m A sequence {Xn : n 1} of X-valued random variables is said to be blockwise m-dependent (resp blockwise and pairwise m-dependent) if for each k ∈ N0 , the collection {Xn : 2k n < 2k+1 } is m-dependent (resp pairwise m-dependent) In the first theorem, we will establish a result on the strong laws of large numbers for sequences of blockwise m-dependent and Cesro CUI X-valued random variables in convex combination space To that, we need the following 15 proposition 2.1.1 Proposition Let K be a compact subset of X If {Xn : n 1} is a sequence of blockwise and pairwise m-dependent X-valued random variables satisfying P (Xn ∈ K) = for all n, then d([n−1 , Xi ]ni=1 , [n−1 , EXi ]ni=1 ) → a.s as n → ∞ 2.1.2 Theorem Let {Xn : n (2.1.1) 1} be a sequence of blockwise m-dependent X-valued random variables and Ces`aro CUI If ∞ n=1 E Xn n2 u0 < ∞, (2.1.2) then (2.1.1) holds For sequences of blockwise and pairwise m-dependent and Ces`aro CUI Xvalued random variables, we obtain the following result 2.1.3 Theorem Let {Xn : n 1} be a sequence of blockwise and pairwise m- dependent X-valued random variables and Ces`aro CUI Suppose that { Xn n u0 : 1} is stochastically dominated by a real-valued random variable X If E |X|(log+ |X|)2 < ∞, (2.1.3) then (2.1.1) holds In the next theorem, we will establish the strong laws of large numbers for sequences of blockwise m-dependent and identically distributed X-valued random variables 2.1.5 Theorem Let {X, Xn : n ≥ 1} be a sequence of blockwise m-dependent and identically distributed X-valued random variables If E X u0 < ∞, (2.1.5) then [n−1 , Xi ]ni=1 → EX a.s as n → ∞ (2.1.6) The following theorem establishes the strong laws of large numbers for sequences of pairwise m-dependent and identically distributed X-valued random variables 16 2.1.6 Theorem Let {X, Xn : n ≥ 1} be a sequence of pairwise m-dependent and identically distributed X-valued random variables Then condition (2.1.5) implies (2.1.6) 2.2 Complete convergence and strong laws of large numbers for triangular arrays of random variables in convex combination space The following proposition is a key to prove a theorem on the complete convergence and the strong laws of large numbers for triangular arrays of random variables in convex combination space 2.2.1 Proposition Let K be a compact subset of X If {Xni : n i 1, n} is a triangular array of rowwise independent X-valued random variables such that P (Xni ∈ K) = for all n 1, i n, then c d([n−1 , Xni ]ni=1 , [n−1 , EXni ]ni=1 ) −→ as n → ∞ and therefore a.s d([n−1 , Xni ]ni=1 , [n−1 , EXni ]ni=1 ) −→ as n → ∞ Applying Proposition 2.2.1, we obtain a result on the complete convergence and the strong laws of large numbers for triangular arrays of rowwise independent and CUI X-valued random variables 2.2.2 Theorem Let {Xni : n 1, i n} be a triangular array of rowwise independent and CUI X-valued random variables Then c d([n−1 , Xni ]ni=1 , [n−1 , EXni ]ni=1 ) −→ as n → ∞ and therefore a.s d([n−1 , Xni ]ni=1 , [n−1 , EXni ]ni=1 ) −→ as n → ∞, if one of the following two conditions are satisfied: (a) The triangular array of real-valued random variables { Xni i u0 :n 1, n} is uniformly bounded by a real-valued random variable X with EX < ∞; (b) For each n 1, the collection { Xni u0 :1 i n} of real-valued ran- dom variables is uniformly bounded by a real-valued random variable Xn with ∞ −p 2p n=1 n E|Xn | < ∞ for some p The following two examples will show that condition (a) in Theorem 2.2.2 is different from (b), even in the case that the convex combination space X = R 17 ∞ 2.2.6 Example Set A = dx x3 (log x)2 Let {X, Xni : n 1, i n} be a collection of independent identically distributed real-valued random variables such that random variable X has probability density function   if x 2, f (x) = Ax3 (log x)2 0 if x < We have EX < ∞ and for all ε > 0, then E|X|2+ε = ∞ Therefore, ∞ n=1 E|Xn |2p = np ∞ n=1 E|X|2p = ∞ for all p np Thus, condition (a) is satisfied but condition (b) is not satisfied 2.2.7 Example Let {Xni : n 1, i n} be a collection of row- wise independent identically distributed real-valued random variables such that: X11 = 0, P (Xn1 = log n) = log−1 n, P (Xn1 = 0) = − log−1 n, for all n Then, we have ∞ n=1 E|Xn1 |4 = n2 ∞ n=1 log3 n < ∞ n2 = log n → ∞ as Thus, condition (b) holds (when p = 2) However, EXn1 n → ∞, so the collection {Xni : n 1, i n} is not uniformly bounded by any square-integrable random variable The conclusions of Chapter In this chapter, we obtain some main results: - Establish some results of the strong laws of large numbers for sequences of random variables in convex combination space satisfying the condition: blockwise and pairwise m-dependent and Ces`aro CUI, or blockwise m-dependent and identically distributed - Establish the complete convergence and the strong laws of large numbers for triangular arrays of rowwise independent and CUI random variables in convex combination space - Give some illustrative examples 18 CHAPTER SOME STRONG LAWS OF LARGE NUMBERS FOR SEQUENCES, TRIANGULAR ARRAYS AND DOUBLE ARRAYS OF FUZZY RANDOM VARIABLES IN CONVEX COMBINATION SPACE In this chapter, first we introduce concept of (α, α+ )-levelwise CUI for collection of fuzzy random variables in convex combination space and concept of (α, α+ )-levelwise Ces`aro r-th CUI for sequence of of fuzzy random variables in convex combination space Next we establish the complete convergence and the strong laws of large numbers for triangular arrays, some results on the strong laws of large numbers for double arrays of fuzzy random variables in convex combination space Finally we present some results on mean convergence and weak laws of large numbers for sequences of fuzzy random variables in convex combination space 3.1 Concepts of (α, α+ )-levelwise CUI and (α, α+ )-levelwise Ces` aro r-th CUI for a collection of fuzzy random variables in convex combination space 3.1.1 Definition A collection {Xi : i ∈ I} of fuzzy random variables in convex combination space X is said to be (α, α+ )-levelwise CUI if {Lα Xi : i ∈ I} is c(X)-valued CUI for each α ∈ (0; 1] and {L+ α Xi : i ∈ I} is c(X)-valued CUI for each α ∈ [0; 1) 3.1.2 Definition A sequence {Xn : n 1} of fuzzy random variables in convex combination space X is said to be (α, α+ )-levelwise Ces`aro r-th CUI if {Lα Xn : n {L+ α Xn : n 3.2 1} is c(X)-valued Ces`aro r-th CUI for each α ∈ (0; 1] and 1} is c(X)-valued Ces`aro r-th CUI for each α ∈ [0; 1) Complete convergence and strong laws of large numbers for triangular arrays of fuzzy random variables in convex combination space 19 The following theorem establishes the complete convergence and the strong laws of large numbers for triangular arrays of rowwise independent and (α, α+ )levelwise CUI fuzzy random variables in the convex combination space 3.2.1 Theorem Let {Xni : n 1, i n} be a triangular array of rowwise independent and (α, α+ )-levelwise CUI fuzzy random variables in convex combination space X, for each ε > 0, there exists a partition = α0 < α1 < · · · < αm = of [0; 1] such that for all n −1 n −1 n max dH L+ αk−1 [n , EF(X) Xni ]i=1 , Lαk [n , EF(X) Xni ]i=1 < ε k m (3.2.1) Then c d∞ [n−1 , Xni ]ni=1 , [n−1 , EF(X) Xni ]ni=1 −→ as n → ∞ and therefore a.s d∞ [n−1 , Xni ]ni=1 , [n−1 , EF(X) Xni ]ni=1 −→ as n → ∞, if one of the following two conditions are satisfied: (a) The triangular array { Xni ∞ :n 1, i n} is uniformly bounded by a real-valued random variable X with EX < ∞; (b) For each n 1, the collection { Xni ∞ by a real-valued random variable Xn with p :1 i n} is uniformly bounded ∞ −p 2p n=1 n E|Xn | < ∞ for some 3.2.2 Corollary Let {Xni : n 1, i n} be a triangular array of row- wise independent and (α, α+ )-levelwise CUI fuzzy random variables in convex combination space X, for each ε > 0, there exists a partition = α0 < α1 < · · · < αm = of [0; 1] such that for all n 1, i n max EdH L+ αk−1 Xni , Lαk Xni < ε k m Suppose that the collection { Xni ∞ :n 1, i n} is uniformly bounded by a real-valued random variable X with EX < ∞ Then c d∞ [n−1 , Xni ]ni=1 , [n−1 , EF(X) Xni ]ni=1 −→ as n → ∞ and therefore a.s (3.2.2) d∞ [n−1 , Xni ]ni=1 , [n−1 , EF(X) Xni ]ni=1 −→ as n → ∞ 20 The following example will show that condition (3.2.1) is actually weaker than (3.2.2) It is considered in the case of X = R, the convex combination operation coincides with the usual addition and scalar multiplication, u0 ≡ 3.2.4 Example For each n 2, we define the functions  |x|   if |x| −  n − 1 un : R → [0; 1], x → − (n − 1)(1 − |x|) if − < |x|   n  0 if |x| > Then, we have un ∈ F(R) for all n Let {Xni : n 1, i , n 1, n} be a triangular array of rowwise independent fuzzy random variables in R such that X11 = I{0} , Xni = ui for all n i 2, n Therefore, Ec(R) Lα Xni = coLα ui = [−1; 1], for α ∈ (0; 1], + Ec(R) L+ α Xni = coLα ui = [−1; 1], for α ∈ [0; 1) Suppose that = α0 < α1 < · · · < αm = is an arbitrary finite partition of [0; 1] From the above arguments, we obtain dH (Ec(R) L+ αk−1 Xni , Ec(R) Lαk Xni ) = dH ([−1; 1], [−1; 1]) = for all k = 1, , m This implies that −1 n −1 n max dH (L+ αk−1 [n , EF(R) Xni ]i=1 , Lαk [n , EF(R) Xni ]i=1 ) k m max k mn n dH (Ec(R) L+ αk−1 Xni , Ec(R) Lαk Xni ) = for all n i=1 Therefore (3.2.1) holds On the other hand, there exists a positive integer number n0 (which depends on the partition above) such that α1 > 1/n for all n n0 For n n0 , we have − α1 − α1 ∪ 1− ;1 n−1 n−1 − α1 It follows that dH (L+ X , L X ) = − → as n → ∞ Thus, there nn α nn n−1 exists ε = 1/2 such that for an arbitrary finite partition of [0; 1], we have L+ Xnn = [−1; 1], Lα1 Xnn = − 1; −1 + max EdH (L+ αk−1 Xnn , Lαk Xnn ) k m EdH (L+ Xnn , Lα1 Xnn ) > ε = 21 when n is sufficiently large Which means that (3.2.2) does not hold 3.3 Strong laws of large numbers for double arrays of fuzzy random variables in convex combination space In the following theorem, we establish the strong laws of large numbers for double arrays of independent and (α, α+ )-levelwise CUI fuzzy random variables in convex combination space 3.3.4 Theorem Let {Xij : i 1, j 1} be a double array of fuzzy random variables in convex combination space X, (α, α+ )-levelwise CUI and for each ε > 0, there exists a finite partition = α0 < α1 < · · · < αp = of [0; 1] such that for all m 1, n 1, we have −1 −1 n m max dH (L+ αk−1 [m , [n , EF(X) Xij ]j=1 ]i=1 , k p Lαk [m−1 , [n−1 , EF(X) Xij ]nj=1 ]m i=1 ) < ε (3.3.1) Then we obtain the strong laws of large numbers a.s −1 −1 n m d∞ ([m−1 , [n−1 Xij ]nj=1 ]m i=1 , [m , [n , EF(X) Xij ]j=1 ]i=1 ) −→ as m ∨ n → ∞, if one of the following two conditions are satisfied: (a) {Xij : i 1, j 1} is a double array of independent fuzzy random variables in convex combination space X, { Xij ∞ :i 1, j 1} is a double array of real-valued random variables which is stochastically dominated by a real-valued random variable X such that E |X| log+ |X| < ∞ (b) {Xij : i 1, j 1} is a double array of independent fuzzy random variables in convex combination space X and ∞ ∞ i=1 j=1 E Xij (ij)r r ∞ < ∞, for some r ∈ [1; 2] In the next theorem, we establish the strong laws of large numbers for double arrays of pairwise independent and identically distributed fuzzy random in convex combination space 3.3.7 Theorem Let {X, Xij : i 1, j 1} be a double array of pairwise in- dependent and identically distributed fuzzy random in convex combination space X If E( X ∞ log + X ∞) < ∞, (3.3.6) 22 then a.s d∞ ([m−1 , [n−1 Xij ]nj=1 ]m i=1 , EF(X) X) −→ as m ∨ n → ∞ 3.4 Mean convergence and weak laws of large numbers for sequences of fuzzy random variables in convex combination space In the first theorem, we establish mean convergence conditions and weak laws of large numbers for sequences of pairwise independent and (α, α+ )-levelwise Ces`aro r-th CUI (r 1) fuzzy random variables in convex combination 3.4.2 Theorem Let {Xn : n 1} be a sequence of pairwise independent and (α, α+ )-levelwise Ces` aro r-th CUI (r 1) fuzzy random variables in convex combination X and for each ε > 0, there exists a finite partition = α0 < α1 < · · · < αp = of [0; 1] such that for all n −1 n −1 n max dH (L+ αk−1 [n , EF(X) Xi ]i=1 , Lαk [n , EF(X) Xi ]i=1 ) < ε k p (3.4.1) Then L r d∞ ([n−1 , Xi ]ni=1 ], [n−1 , EF(X) Xi ]ni=1 ) −→ as n → ∞ and therefore P dr∞ ([n−1 , Xi ]ni=1 ], [n−1 , EF(X) Xi ]ni=1 ) −→ as n → ∞ In the next part, we establish necessary and sufficient conditions for mean convergence of sequences of fuzzy random variables in convex combination space without independent hypothesis and compactly uniformly integrable hypothesis 3.4.3 Theorem Let {Xn : n 1} be a sequence of fuzzy random variables in convex combination space X and for each ε > 0, there exists a finite partition = α0 < α1 < · · · < αp = of [0; 1] such that for all n −1 n −1 n E max dH (L+ αk−1 [n , Xi ]i=1 , Lαk [n , Xi ]i=1 ) < ε k p Then L d∞ ([n−1 , Xi ]ni=1 , [n−1 , EF(X) Xi ]ni=1 ) −→ as n → ∞ if and only if for each α ∈ (0; 1] L dH ([n−1 , Lα Xi ]ni=1 , [n−1 , Ec(X) Lα Xi ]ni=1 ) −→ as n → ∞ (3.4.2) 23 In the case, the sequence of expectations of fuzzy random variables in convex combination space is convergent with respect to metric d∞ , we obtain the following result 3.4.5 Theorem Let {Xn : n 1} be a sequence of fuzzy random variables in convex combination space X and there exists v ∈ F(X) such that d∞ ([n−1 , EF(X) Xi ]ni=1 , v) → as n → ∞ (3.4.4) Then L d∞ ([n−1 , Xi ]ni=1 , [n−1 , EF(X) Xi ]ni=1 ) −→ as n → ∞ if and only if for each α ∈ (0; 1] L dH ([n−1 , Lα Xi ]ni=1 , [n−1 , Ec(X) Lα Xi ]ni=1 ) −→ as n → ∞ and for each α ∈ [0; 1) L n −1 + n dH ([n−1 , L+ α Xi ]i=1 , [n , Ec(X) Lα Xi ]i=1 ) −→ as n → ∞ The conclusions of Chapter In this chapter, we obtain some main results: - Establish the complete convergence and the strong law of large numbers for triangular arrays of rowwise independent and (α, α+ )-levelwise CUI fuzzy random variables in convex combination space - Establish some results on the strong law of large numbers for double arrays of fuzzy random variables in convex combination space satisfying the condition: independent and (α, α+ )-levelwise CUI, or pairwise independent and identically distributed - Establish the mean convergence and weak law of large numbers for sequences of pairwise independent and (α, α+ )-levelwise Ces`aro r-th CUI (r 1) fuzzy random variables in convex combination space Establish necessary and sufficient conditions on the mean convergence for sequences of fuzzy random variables in convex combination space - Give some illustrative examples 24 GENERAL CONCLUSIONS AND SUGGESTIONS General conclusions In this thesis, we obtain some main results: - Establish some results on the strong law of large numbers for sequences of random variables in convex combination space satisfying the condition: blockwise and pairwise m-dependent and Ces`aro CUI, or blockwise m-dependent and identically distributed - Establish the complete convergence and the strong law of large numbers for triangular arrays of rowwise independent and CUI random variables in convex combination space - Establish the complete convergence and the strong law of large numbers for triangular arrays of rowwise independent and (α, α+ )-levelwise CUI fuzzy random variables in convex combination space - Establish some results on the strong law of large numbers for double arrays of fuzzy random variables in convex combination space satisfying the condition: independent and (α, α+ )-levelwise CUI, or pairwise independent and identically distributed - Establish the mean convergence and weak law of large numbers for sequences of pairwise independent and (α, α+ )-levelwise Ces`aro r-th CUI (r 1) fuzzy random variables in convex combination space Establish necessary and sufficient conditions on the mean convergence for sequences of fuzzy random variables in convex combination space Recommendations In the near future we will study the following issues: - Weak law of large numbers for sequences and arrays of random variables in convex combination space - Law of large numbers on Wijsman convergence for sequences and arrays of random variables in convex combination space 25 LIST OF THE AUTHOR’S ARTICLES RELATED TO THE THESIS [1] N T Thuan, N V Quang and P T Nguyen (2014), On complete convergence for arrays of rowwise independent of random variables in convex combination spaces, Fuzzy Sets and Systems, 250, 52-68 [2] N V Quang and P T Nguyen (2015), Some strong laws of large numbers for double array of random upper semicontinuous functions in convex combination spaces, Statistics and Probability Letters, 96, 85-94 [3] N V Quang and P T Nguyen (2016), Strong Law of Large Numbers for Sequences of Blockwise and Pairwise m-Dependent Variables in Metric Spaces, Applications of Mathematics, 6, 669-684 Results of the thesis have been reported in: - The 8th Vietnamese Mathematical Conference (University of Information Officers, 10-14/08/2013), - The 5th National Conference: ”Probability - Statistics: Research, Application and Teaching” (Da Nang University of Education, 23-25/05/2015), - Seminar of Department of Probability - Statistics and Application, Institute of Science Education, Vinh University (from 2012 to 2016)  ... Colubi (in 1999) derived the strong law of large numbers for independent and identically distributed fuzzy random variables in Rd Proske and Puri (in 2002) proved the strong law of large numbers... establish the strong law of large numbers for sequences of random variables and arrays of fuzzy random variables in convex combination space, to establish complete convergence and strong law of... Subject of the research - The strong law of large numbers for sequences of random variables, double arrays of fuzzy random variables - The complete convergence and strong law of large numbers for