SOME LAWS OF LARGE NUMBERS FOR MULTIDIMENSIONAL ARRAYS AND TRIANGULAR ARRAYS OF RANDOM SETS.

SOME LAWS OF LARGE NUMBERS FOR MULTIDIMENSIONAL ARRAYS AND TRIANGULAR ARRAYS OF RANDOM SETS.SOME LAWS OF LARGE NUMBERS FOR MULTIDIMENSIONAL ARRAYS AND TRIANGULAR ARRAYS OF RANDOM SETS.SOME LAWS OF LARGE NUMBERS FOR MULTIDIMENSIONAL ARRAYS AND TRIANGULAR ARRAYS OF RANDOM SETS.SOME LAWS OF LARGE NUMBERS FOR MULTIDIMENSIONAL ARRAYS AND TRIANGULAR ARRAYS OF RANDOM SETS.SOME LAWS OF LARGE NUMBERS FOR MULTIDIMENSIONAL ARRAYS AND TRIANGULAR ARRAYS OF RANDOM SETS.SOME LAWS OF LARGE NUMBERS FOR MULTIDIMENSIONAL ARRAYS AND TRIANGULAR ARRAYS OF RANDOM SETS.SOME LAWS OF LARGE NUMBERS FOR MULTIDIMENSIONAL ARRAYS AND TRIANGULAR ARRAYS OF RANDOM SETS.

MINISTRY OF EDUCATION AND TRAINING VINH UNIVERSITY -* - BUI NGUYEN TRAM NGOC SOME LAWS OF LARGE NUMBERS FOR MULTIDIMENSIONAL ARRAYS AND TRIANGULAR ARRAYS OF RANDOM SETS Speciality: Theory of probability and mathematical Statistics Code: 9460106 A SUMMARY OF MATHEMATICS DOCTORAL THESIS NGHE AN - 2022 Work is completed at Vinh University Supervisor: Prof Dr Nguyen Van Quang Dr Duong Xuan Giap Reviewer 1: Reviewer 2: Reviewer 3: Thesis will be defended at school-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 an important role in the development of probability theory They have been extensively studied and applied in several fields, such as optimization and control, stochastic and integral geometry, mathematical economics, statistics and related fields 1.2 In the last 40 to 50 years, one of the directions in studying the limit theorems in probability theory is to extend the results for single-valued random variables to set-valued random variables (random sets) This research can be applied in several fields such as optimization and control, stochastic and integral geometry, mathematical economics, etc However, since the space of closed subsets of Banach space does not have the structure of a vector space, there are several irregularities in the study and establishment of limit theorems Therefore, the study of numerical law for random sets is not only theoretical but also practical 1.3 For multi-indexed structure, the usual partial order relation is not complete So, if we extend the limit theorems for random sets from the sequence case to the multidimensional array case, then we will have a lot of news thing This implies the results of multi-valued laws of large numbers more interesting 1.4.In Vietnam, the limit theorems for single-valued random variables vector space, there are several irregularities in the study and establishment of limit theorems For the random sets case, in the last 10 years, some interesting results have been introduced by Nguyen Van Quang, Duong Xuan Giap, Nguyen Tran Thuan, Hoang Thi Duyen, However, there are still many other results for the single-valued random variables case that have not been extended to the random sets case So, there will be many interesting issues to study if we extend the results from arrays of single-valued random variables case to arrays of random sets case With the above reasons, we have chosen the topic for the thesis as follows: “Some laws of large numbers for multidimensional arrays and triangular arrays of random sets” Objective of the research The research subjects of the thesis are to establish some laws of large numbers for multidimensional arrays and triangular arrays of random sets under different conditions Subject of the research The research subject of the thesis is the random sets, the random upper semicontinuous functions and some dependencies of random sets such as: pairwise independent, uniformly integrable compact, negative dependence, negative association Scope of the research The thesis focuses on studying the laws of large numbers for double arrays and triangularly arrays of random sets with gap topology Additionally, the thesis also establishes some laws of large numbers for d-dimensional arrays of random upper semicontinuous functions Methodology of the research We use a combination of the fundamental methods of probability theory, convex analysis and functional analysis such as: the convexification technique, the blocking procedure in proving the law of large numbers Contributions of the thesis The results of the thesis help to expand the research direction of the limit theorems for random sets The thesis can be used as a reference for students, graduate students and PhD students majoring in Theory of probability and mathematical Statistics Overview and organization of the research 7.1 Overview of the research In this thesis, we extend the P Ter´an’s results for double arrays of random sets and combining P Ter´an’s method with the techniques for building a double array of selections developed by Nguyen Van Quang and his fellows to prove the “liminf” part of Painlev´eKuratowski con- vergence Using these results, we establish some laws of large numbers for double arrays of random sets with the gap topology for the case m ∨ n → ∞ For the triangular arrays of random sets, we establish some strong laws of large numbers of rowwise independent random sets, compactly uniformly integrable and satisfying some various conditions To this, we present some strong laws of large numbers for triangular arrays of random elements and prove the “liminf” part of Painlev´e-Kuratowski convergence Finally, by extending the results that Nguyen Van Quang and Duong Xuan Giap introduced in 2013, we establish the strong law of large numbers for triangular arrays of random sets taking closed- values of Rademacher type p Banach space Extending some results that Nguyen Tran Thuan and Nguyen Van Quang introduced in 2016 for the case multidimensional arrays, we obtain some laws of large numbers for negatively associated and pairwise negatively dependent random upper semicontinuous functions To prove these limit theorems, we also introduce H´ajek-R´enyi’s type maximal inequality for an array of negatively associated random upper semicontinuous functions and the law of large numbers for d-dimensional arrays of pairwise negatively dependent realvalued random variables 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 and appendix Chapter introduces some preliminaries Chapter presents some strong laws of large numbers for double arrays of random sets with gap topology Chapter establishes some strong laws of large numbers for triangular arrays of of rowwise independent random sets with gap topology Appendix provides some maximal inequalities which form Rosenthal’s type and Ha ´jek- R´enyi’s type for multi-dimensional structure and establishes some laws of large numbers for d-dimensional arrays of level-wise negatively associated and level-wise pairwise negatively de- pendent random upper semicontinuous functions under various settings CHAPTER PRELIMINARIES In this chapter, we introduce some important types of convergence on the space of closed subsets of Banach space, some preliminaries of random sets and present some basic notions and related properties for random upper semicontinuous functions 1.1.The convergence on the space of closed subsets of Banach space Suppose that t is a topology on X, {An : n ∈ Nd} is an array on the space of closed subsets, nonempty c(X) of X We put t- lim inf An = {x ∈ X : x = tlim xn, xn ∈ An, n ∈ Nd}, t- lim sup An = {x ∈ X : x = tlim xk, xk ∈ Ank |n|→∞ |n|→∞ |k|→∞ |n|→∞ , k ∈ Nd}, where {Ank : k ∈ Nd} is a sub-array of {A : n ∈ Nd} The sets t- lim inf A and t- lim sup A n n n |n|→∞ |n|→∞ d are lower limit and upper limit of the array {An : n ∈ N }, relative to topology t as |n| → ∞ Definition 1.1.6 Let A ∈ c(X) The array {An : n ∈ Nd} ⊂ c(X) is said to be (1) converges in the sense of Mosco to A as |n| → ∞ and is denoted by An (M) −→ A as |n| → ∞ or (M)- lim An = A, if w- lim sup An = s- lim inf An = A | n|→∞ |n|→∞ |n|→∞ (2) converges in the sense of Painlev´e - Kuratowski to A with respect to the strong topology s of X as |n| → ∞ and is denoted by An s- lim sup An = s- lim inf An = A (K) −→ A as |n| → ∞ or (K)- lim |n|→∞ |n|→∞ | An = A, if n|→∞ (W) (3) converges in the sense of Wijsman to A as |n| → ∞ and is denoted by An − → A as |n| → ∞ or (W)- lim | n|→∞ An = A, if lim d(x, An) = d(x, A) for every x ∈ X |n|→∞ A slice of a ball is the intersection of a closed ball B¯ (x, r) (where x ∈ X, r > 0) and of a closed half space F (x∗, α) = {x ∈ X : ⟨x∗, x⟩ ≥ α} (where x∗ ∈ X∗, x∗ ̸= and α ∈ R) (S) (4) converges in the slice topology to A |n| → ∞ and is denoted by An −→ A as |n| → ∞ or (S)- lim An = A, if lim D(An, C) = D(A, C) for all nonempty slices C of balls | n|→∞ |n|→∞ of X (G) (5) converges in the gap topology to A as |n| → ∞ is denoted by An −→ A as |n| → ∞ or (G)- lim |n|→∞ An = A, if lim D(An, C) = D(A, C) for all nonempty bounded closed convex |n|→∞ subsets C of X Remark 1.1.7 (1) (K)lim | An = A if and only if s- lim sup An ⊂ A ⊂ s- lim inf An n|→∞ (M)- |lim n|→∞ |n|→∞ |n|→∞ An = A if and only if w- lim sup An ⊂ A ⊂ s- lim inf An |n|→∞ |n|→∞ (2) On the space of convex compact and non-empty subsets, Wijsman convergence leads to Painlev´e - Kuratowski convergence On the space of convex, closed and non-empty subsets, the types of convergence: Wijsman convergence, slice topology and gap topology are equivalent 1.2.The random sets In this part, we introduce the random sets and some basic notions Let Bc(X) be the σ-field on c(X) generated by the sets U − := {C ∈ c(X) : C ∩ U ̸= ∅}, for all open subsets U of X We call Bc(X) the Effrăos -field Definition 1.2.1 A mapping X : Ω → c(X) is said to be F-measurable if for every B ∈ Bc(X), X−1(B) ∈ F The mapping F-measurable X is also called F-measurable random set If F = A then X is said for shortly to be random set lim sup s(x∗, clEFni) s(x∗, X) for all x∗ X∗, n→∞ (3.12) the n cl n Σ F (ω) → coX n ni i=1 almost surely, in the gap topology, as n → ∞ Theorem 3.2.4 Let {Xni : n ≥ 1, ≤ i ≤ n} be a triangular array of rowwise independent and compactly uniformly integrable random sets in a separable Banach space If {∥Xni∥ : n ≥ 1, ≤ i ≤ n} are uniformly bounded by a random variable ξ with Eξ < ∞ and there exists X ∈ k(X) satisfying (3.11) and (3.12), then cl n Σ F (ω) → coX n ni i=1 almost surely, in the gap topology, as n → ∞ The theorem below will establish the strong laws of large numbers for a triangular array of rowwise independent random sets in a separable Banach space of type p (1 < p ≤ 2) with respect to the gap topology, based on Ter´an’s method and the results of Quang and Giap in 2013 Theorem 3.2.5 Let {Xni : n ≥ 1, ≤ i ≤ n} be a triangular array of rowwise independent, bounded expectation random sets in a separable Banach space of type p (1 < p ≤ 2) Let {an} be a sequence of positive real numbers such that an+1 > an and lim an = +∞ Let ψ(t) be a n→∞ positive, even, convex, continuous function such that ψ(|t|) |t|r of |t| , ψ(|t|) |t|r+p−1 is a monotone increasing function is a monotone decreasing function of |t| for some nonnegative integer r and there exists a positive constant C1 so that ψ(t) satisfying (3.2) If the following conditions are satisfied Σ Σ ∞ n E ψ(∥F n=1 i=1 ∥) ni Σ ψ(n) < ∞; Σ Σn n=1 F Σp.k p ∞ E i= n i n < ∞, for some positive integer k and there exists X ∈ k(X) satisfying (3.11) and (3.12), then n cl Σ F (ω) → coX n ni i=1 almost surely, in the gap topology, as n → ∞ The conclusions of Chapter In this chapter, we obtain some main results: - Establish some strong laws of large numbers for triangular arrays of rowwise independent and compactly uniformly integrable random elements ; - Prove some strong laws of large numbers for triangular arrays of rowwise independent, compactly uniformly integrable random sets and satisfying some different conditions; - Present the strong laws of large numbers for a triangular array of rowwise independent random sets in a separable Banach space of type p APPENDIX THE LAWS OF LARGE NUMBERS FOR ARRAYS OF RANDOM UPPER SEMICONTINUOUS FUNCTIONS An extension to random sets is to study random variables taking fuzzy set values (or, random upper semicontinuous functions) In this part, at first we prove some inequalities and establish some laws of large numbers for arrays of real-valued random variables Later, we introduce some laws of large number for multidimensional arrays of level-wise negatively associated and level-wise pairwise negatively dependent random upper semicontinuous functions Let {bn, n ∈ Nd} be a d-dimensional array of real numbers We define ∆bn to be the d-th order finite difference of the b’s at the point n by ∆bn ΣΣ := d i= (ni −di ) bk k Thus, bn = Σ 1⪯k⪯n (−1) ∆bk for all n ∈ Nd If d = then ∆bi = bi − bi−1; if d = then for all (i, j) ∈ N2 , ∆bij = bij −bi,j−1 −bi−1,j +bi−1,j−1 (with the convention that b0,0 = bi,0 = b0,j = 0) For convenience, from now until the end of this part, random upper semicontinuous functions is said for shortly to be U-valued random variables and we use Sk to denote the k-th partial sum of an array {Xn, n ∈ Nd}, it means Sk = Σ Xi i⪯k 4.1The laws of large number for arrays of level-wise negatively asso- ciated random upper semicontinuous functions At first, we introduce some maximal moment inequalities for arrays of negatively associated real-valued random variables Proposition 4.1.3 Let {Xn, n ∈ Nd} be an array of negatively associated real- valued random variables withn EXn = and EX < ∞ Then there exists a positive constant Cd depending on d such that for every n, E max m⪯ n X i ≤ C d Σ EXi Σ Σ2 1/2 (4.1) i i ⪯m ⪯n Proposition 4.1.4 Let p ≥ and let {Xn, n ∈ Nd} be an array of negatively associated real-valued random variables with EXn = and E|Xn|p < ∞, n ∈ Nd Then there exists a constant Cp,d depending only on p and d such that for every n, we have E max m⪯n p Xi ≤ EXi Σ Cp,d Σ i⪯n Σ2 p/2 + Σ E|Xi| p Σ (4.2) i⪯n i ⪯m The following theorem establishes H´ajek-R´enyi’s type maximal inequality for an array of level-wise negatively associated U-valued random variables and this result is obtained in the setting with respect to metric D∗ This theorem plays the key role to derive the laws of large numbers Theorem 4.1.5 Let {bn, n ∈ Nd} be an array of positive real numbers with ∆bn ≥ for all n ∈ Nd Assume that {Xn, n ∈ Nd} is an array of D∞-integrable, level∗ variables with E∥Xn∥2 < ∞, n ∈ Nd wise negatively associated U -valued random Then, there exists a positive constant Cd depending only on d such that for any ε > and for any points m ⪯ n, k P max m⪯k⪯n bk D∗( S k , ES Σ )≥ε ≤ Cd VarXi (bi + b m) Σ ε2 (4.3) i⪯ n Theorem 4.1.6 Let {bn, n ∈ Nd} be an array of positive real numbers with ∆bn ≥ for all n ∈ Nd and {Xn, n ∈ Nd} be an array of D∞-integrable, level-wise negatively associated U -valued random variables with E∥Xn∥2 < ∞, n ∈ Nd If ∗ nu mb ers then the strong law of large Σ n b ∈Nd VarXn n < ∞, (4.4 ) bn D∗(Sn, ESn) → h.c.c n → ∞ The following theorem establishes the weak law of large numbers for an array of level-wise negatively associated U-valued random variables Theorem 4.1.7 Let {Xn, n ∈ Nd} be an array of D∞-integrable, level-wise negatively associated U -valued random variables with E∥Xn∥2 < ∞, n ∈ Nd ∗ Σ Assume that {bn, n ∈ Nd} be an array of positive real numbers with ∆bn ≥ for all n ∈ Nd If b−n VarXi → as i⪯n n → ∞, then bn max D∗(Sk , ) as n → ∞ k⪯n ESk − →P 4.2 The laws of large number for arrays of level-wise pairwise nega- tively dependent random upper semicontinuous functions The following lemma establishes a strong law of large numbers for d-dimensional arrays of pairwise negatively dependent real-valued random variables In the case of pairwise independent random variables, it is proved by Fazekas and T´om´acs Proposition 4.2.3 Let {Xn, n ∈ Nd} be an array of pairwise negatively dependent real- valued random variables Assume that (1) sup E| Xn| 0, there Σ n(ε) = n1 (ε), , nd (ε) ∈ Nd and a partition = α0 < α1 < · · · < αt = of [0; 1] such that Σ max dH [ESn ]αk−1 + , [ESn ]αk 1≤k≤ t ≤ |n|ε for all n ≥ n(ε) (4.13) If sup E∥ Xn∥∞ < ∞ , then n∈Nd |n| D∞ n , ESn Σ → a.s as |n| → ∞ S technical condition (4.13) to obtain the strong law of large In Theorem 4.2.9, we use the numbers The proposition below will give some sufficient conditions for (4.13) Proposition 4.2.10 The condition (4.13) holds if the array {Xn, n ∈ Nd} satisfies one of the following conditions (1) {Xn, n ∈ Nd} is an array of identically distributed U -valued random variables (2) There exists an element u ∈ U such that [u](1), [u](2) are continuous functions with respect α α to α on [0; 1] and for each α ∈ [0; 1], j = 1, 2, Σ [EXi ](j) → [u](j)α |n| → ∞ |n| α (4.10) i⪯n −1 d (3) {|n| ESn, n ∈ N } is a convergent array with respect to metric D∞ (4) {Xn, n ∈ Nd} is an array D∞-compactly uniformly integrable (5) There exists a compact subset C of (U , D∞) such that EXn ∈ C for all n ∈ Nd The conclusions of Appendix In this appendix, we obtain some main results: - Introduce some maximal moment inequalities for arrays of negatively associated realvalued random variables; - Prove the law of large numbers for d-dimensional arrays of pairwise negatively dependent real-valued random variables; - Establish H´ajek-R´enyi’s type maximal inequality for an array of level-wise negatively associated U-valued random variables and present some laws of large number for ddimensional arrays of level-wise negatively associated and level-wise pairwise negatively dependent random upper semicontinuous functions GENERAL CONCLUSIONS AND SUGGESTIONS General conclusions In this thesis, we obtain some main results: - Establish some strong laws of large numbers for double arrays of random sets for the cases: pairwise independent and compactly uniformly integrable in the Ces`aro sense, or pairwise independent and identically distributed, or independent and taking closed-values of Rademacher type p Banach space - Establish some strong laws of large numbers for triangular arrays of rowwise independent random sets - Prove some maximal moment inequalities for d-dimensional arrays of negatively associated real-valued random variables and introduce the law of large numbers for d-dimensional arrays of pairwise negatively dependent real-valued random variables - Establish H´ajek-R´enyi’s type maximal inequality for an array of level-wise negatively associated U-valued random variables and present some laws of large number for ddimensional arrays of level-wise negatively associated and level-wise pairwise negatively dependent random upper semicontinuous functions Recommendations In the near future we will study the following issues: - Establish some laws of large number for triangular arrays of rowwise level-wise negatively associated and rowwise level-wise negatively dependent random upper semicontinuous functions - Establish some laws of large number for d-dimensional arrays of level-wise m-dependence random upper semicontinuous functions LIST OF THE AUTHOR’S ARTICLES RELATED TO THE THESIS N V Quang, D X Giap, B N T Ngoc, T C Hu (2019), Some strong laws of large numbers for double arrays of random sets with gap topology, Journal of Convex Analysis, 26, 719-738 D X Giap, N V Huan, B N T Ngoc, N V Quang (2020), Multivalued strong laws of large numbers for triangular arrays with gap topology, Journal of Convex Analysis, 27, 1157 - 1176 D X Giap, N V Quang, B N T Ngoc (2021), Some laws of large numbers for arrays of random upper semicontinuous functions, Fuzzy Sets and Systems, https://doi.org/10.1016/j.fss.2021.08.015, In Press D X Giap, N V Quang, B N T Ngoc (2021), Some laws of large numbers for double arrays of random upper semicontinuous functions, Vinh University Journal of science, Vol.50-2A (2021), 26 - 40 Results of the thesis have been reported in: - The 2th Mien Trung-Tay Nguyen Mathematics Conference (Da Lat University, 0911/12/2017); - 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); - The 6th National Conference “ Probability - Statistics: Research, Application and Teaching” (Can Tho University, 05-08/11/2020); - Seminar of Department of Probability - Statistics and Application, Institute of Science Education, Vinh University (from 2016 to 2020) ... establish some laws of large numbers for double arrays of random sets with the gap topology for the case m ∨ n → ∞ For the triangular arrays of random sets, we establish some strong laws of large numbers. .. chapter, at first, we prove some strong laws of large numbers of triangular arrays of random elements Next we establish some strong laws of large numbers of triangular arrays of random sets with respect... topology 3.1.The strong laws of large numbers of triangular arrays of random elements In this section, we will establish some strong laws of large numbers of triangular arrays of random elements under