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

NGHE AN - 2022

Work is completed at Vinh University

Supervisor: 1 Prof Dr Nguyen Van Quang

2 Dr Duong Xuan Giap

at , ,

Thesis can be found at:

- Vietnam National Library

- Nguyen Thuc Hao Library and Information Center - Vinh University

PREFACE

1 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

proba-bility theory is to extend the results for single-valued random variables to set-valued randomvariables (random sets) This research can be applied in several fields such as optimization andcontrol, stochastic and integral geometry, mathematical economics, etc However, since thespace of closed subsets of Banach space does not have the structure of a vector space, there areseveral irregularities in the study and establishment of limit theorems Therefore, the study ofnumerical 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 multidimensionalarray 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 setscase, 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 manyother results for the single-valued random variables case that have not been extended to therandom sets case So, there will be many interesting issues to study if we extend the resultsfrom arrays of single-valued random variables case to arrays of random sets case

2With 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”.

2 Objective of the research

The research subjects of the thesis are to establish some laws of large numbers for dimensional arrays and triangular arrays of random sets under different conditions

multi-3 Subject of the research

The research subject of the thesis is the random sets, the random upper semicontinuousfunctions and some dependencies of random sets such as: pairwise independent, uniformlyintegrable compact, negative dependence, negative association

4 Scope of the research

The thesis focuses on studying the laws of large numbers for double arrays andtriangularly arrays of random sets with gap topology Additionally, the thesis also establishessome laws of large numbers for d-dimensional arrays of random upper semicontinuousfunctions

5 Methodology of the research

We use a combination of the fundamental methods of probability theory, convex analysisand functional analysis such as: the convexification technique, the blocking procedure inproving the law of large numbers

6 Contributions of the thesis

The results of the thesis help to expand the research direction of the limit theorems forrandom sets

The thesis can be used as a reference for students, graduate students and PhD studentsmajoring in Theory of probability and mathematical Statistics

7 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 bining P Ter´an’s method with the techniques for building a double array of selectionsdeveloped by Nguyen Van Quang and his fellows to prove the “liminf” part of Painlev´e-Kuratowski con- vergence Using these results, we establish some laws of large numbers fordouble arrays of random sets with the gap topology for the case m ∨ n → ∞.

com-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 somevarious conditions To do this, we present some strong laws of large numbers for triangulararrays 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 takingclosed- 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 fornegatively associated and pairwise negatively dependent random upper semicontinuousfunctions To prove these limit theorems, we also introduce H´ajek-R´enyi’s type maximalinequality for an array of negatively associated random upper semicontinuous functions andthe law of large numbers for d-dimensional arrays of pairwise negatively dependent real-

valued random variables

7.2 The organization of the research

Besides the sections of usual notations, preface, general conclusions, andrecommendations, list of the author’s articles related to the thesis and references, the thesis isorganized into three chapters and appendix

Chapter 1 introduces some preliminaries

Chapter 2 presents some strong laws of large numbers for double arrays of random setswith gap topology

Chapter 3 establishes some strong laws of large numbers for triangular arrays of of rowwise

5independent 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 largenumbers for d-dimensional arrays of level-wise negatively associated and level-wise pairwise

negatively de- pendent random upper semicontinuous functions under various settings

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 closedsubsets, nonempty c(X) of X We put

are lower limit and upper limit of the array {An : n ∈ Nd}, 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)

(3) converges in the sense of Wijsman to A as |n| → ∞ and is denoted by An − → A as

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∗ ̸= 0 and α ∈ R)

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 Effr¨os σ-field.

Definition 1.2.1 A mapping X : Ω → c(X) is said to be F-measurable if for every B B∈ c(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.

Let X be a random set, we define A X = {X−1(B), B B∈ c(X)} Then AX is the smallest

sub σ-field of A with respect to X measurable Distribution of X is a probability measure P X

on Bc(X) defined by

PX (B) = P.X−1(B)Σ, B ∈ B c(X)

Definition 1.2.2 A family of random sets {X i , i ∈ I} is said to be independent

(respectively, pairwise independent ) if {A X i , i ∈ I} are independentl (respectively,

pairwise independent), and is said to be identically distributed if all P X i , i ∈ I are

) p is finite If F = A then Lp(F, X) is denoted

for shortly by Lp(X) If X = R then Lp replace L p(R)

For each random set F-measurable X and for p ≥ 1, we denote S p (F) = {f ∈ Lp(F, X) :

f (ω) ∈ X(ω) a.s.} In the case F = A then S p (A) is replaced by S p

Definition 1.2.4. The random set X : Ω → c(X) is called

Definition 1.2.5.The expectation of integrable random set X, denoted by EX, is defined

by

EX := {Ef : f ∈ S1 }where Ef is the usual Bochner integral of random element f

In Definition 1.1.6, if we replace An by Xn(ω) and replace A by X(ω) for ω in a set with

probability 1, where X, Xn, n ∈ Nd are random sets, then we obtain the definition of almostsure convergence for random sets

1.3.The uniformly integrable compactness and the uniformly bounded of an array of random sets

We present some notions: compactly uniformly integrable, compactly uniformly integrable

in the Ces`aro sense, uniformly bounded for double array and triangular array of random

ele-ments and random sets.

Definition 1.3.1. (1) A double array of random elements {f mn : m ≥ 1, n ≥ 1} is said to

be

uniformly integrable in the Ces`aro sense if

(2) A double array of random elements {f mn : m ≥ 1, n ≥ 1} is said to be compactly

uniformly integrable in the Ces`aro sense if for every ε > 0, there exists a compact subset K

Σ < ε.

In general, compactly uniform integrability in the Ces`aro sense is stronger than theuniform integrability in the Ces`aro sense, but they are equivalent in the real or the identicallydistributed case

Let A be a subset of Ω, the complement of the set A with respect to Ω is denoted by A c.(3) A double array of random sets {X mn : m ≥ 1, n ≥ 1} is said to be compactly uniformly integrable in the Ces`aro sense if for every ε > 0, there exists a compact subset K of X such

(.)⊂ K) c

Σ < ε.

Definition 1.3.2. (1) A triangular array of random elements {f ni : n ≥ 1, 1 ≤ i ≤ n} is

said to be compactly uniformly integrable if for every ε > 0, there exists a compact subset K

(2) A triangular array of random sets {X ni : n ≥ 1, 1 ≤ i ≤ n} is said to be compactly

uniformly integrable if for every ε > 0, there exists a compact subset K of X such that

n, i

Definition 1.3.3 (1) A triangular array of random elements {f ni : n ≥ 1, 1 ≤ i ≤ n} is

uniformly bounded by a random variable ξ if for all n ≥ 1, 1 ≤ i ≤ n and for every real number

t > 0,

P (∥f ni∥> t) ≤ P (|ξ| > t).

(2) A triangular array of random sets {X ni : n ≥ 1, 1 ≤ i ≤ n} is uniformly bounded by a

random variable ξ if for all n ≥ 1, 1 ≤ i ≤ n and for every real number t > 0,

P (∥X ni∥> t) ≤ P (|ξ| > t).

1.4.The random upper semicontinuous functions

In this section, we sum up some basic notions and related properties for random uppersemicontinuous functions

Let K be the set of compact intervals of R If x is an element of K, then it will be denoted

by x = [x(1); x(2)], where x(1), x(2) are two end points The Hausdorff distance d H on K is defined

by

d H (x, y) = max{|x(1) − y(1)|; |x(2) − y(2)|}, x, y K∈ .

For a function u : R → [0; 1], the α-level set of u is defined by [u] α = {x ∈ R : u(x) ≥ α}

for each α ∈ (0; 1] For each α ∈ [0; 1), [u] α+ denotes the closure of {x ∈ R : u(x) > α}.

In particular, [u]0+ is called the support of u and denoted by supp u The level set [u] α of

u is closed for all α ∈ (0; 1] iff u is upper semicontinuous function A upper semicontinuous

function u : R → [0; 1] is called quasiconcave function if u(λx x + (1 − λx )y) ≥ min{u(x), u(y)}

for all x, y ∈ R, and its equivalent condition is that [u] α is a convex subset of R for every

α ∈ (0; 1] Let U denote the family of all upper semicontinuous functions u : R → [0; 1]satisfying the following conditions

(1) supp u is compact;

(2) [u]1 ̸= ∅};

, where [u](1) and [u](2) are two end points of the interval.

Definition 1.4.2 A mapping X : Ω → K is called a K-valued random variable if X−1(B) ∈

A

for all B B∈ (K), where B(K) is the Borel σ-algebra on (K, d H ).

Definition 1.4.3 A mapping X : Ω → U is called a U-valued random variable (or fuzzy random variable, or random upper semicontinuous function) if [X] α is a K-valued random

variable for all α ∈ (0; 1]

The following is the concepts of negative dependence and negative association for the casemultidimensional

Definition 1.4.9 (1) A finite family {X i , 1 ≤ i ≤ n} of real-valued random variables is

said to be negatively associated if for any disjoint subsets A1, A2 of {1, 2, , n} and any

real coordinatewise nondecreasing functions f on R|A1 |, g on R|A2 | such that

CovΣ

f1(X i , i ∈ A1), f2(X j , j ∈ A2)Σ

≤ 0,

whenever the covariance exists, where |A| denotes the cardinality of A.

An infinite collection of real-valued random variables is negatively associated if every finitesubfamily is negatively associated

(2) Let {Xn, n ∈ Nd} be an array of K-valued random variables Then, {Xn, n ∈ Nd} issaid to be negatively associated if {X(1), n ∈ Nd} and {X(2), n ∈ Nd} are arrays of negativelyassociated real-valued random variables

(3) Let {Xn, n ∈ Nd} be an array of U-valued random variables Then {Xn, n ∈ Nd} issaid to be level-wise negatively associated if {[Xn]α , n ∈ Nd} are arrays of negativelyassociated K-valued random variables for all α ∈ (0; 1]

Definition 1.4.10 (1) A finite family {X i , 1 ≤ i ≤ n} of real-valued random variables is

Y

n n

said to be negatively dependent if the two following inequalities hold

n

P(X1 > x1, , X n > x n) ≤ P(X i > x i),

i=1 n

A family {X i , i ∈ I} of real-valued random variables is pairwise negatively

dependent if P(X i > x, X j > y) ≤ P(X i > x)P(X j > y) (or equivalently, P(X i ≤ x, X j ≤ y)

≤ P(X i ≤ x)P(X j ≤ y)) for all i ̸= j and all x, y ∈ R

(2) Let {Xn, n ∈ Nd} be an array of K-valued random variables Then, {Xn, n ∈ Nd}

is said to be negatively dependent (resp pairwise negatively dependent ) if {X(1), n ∈

Nd} and {X(2), n ∈ Nd} are arrays of negatively dependent (resp pairwise negatively dependent)real-valued random variables

(3) Let {Xn, n ∈ Nd} be an array of U-valued random variables Then {Xn, n ∈ Nd}

is said to be level-wise negatively dependent (resp level-wise pairwise negatively dependent ) if {[Xn]α , n ∈ Nd} are arrays of negatively dependent (resp pairwise negativelydependent) K-valued random variables for all α ∈ (0; 1]

The conclusions of Chapter 1

In this chapter, we obtain some main results:

- Introduce some important convergences on the space of closed subsets of Banach space;

- Present random sets and some and some related concepts;

- Present some basic notions and related properties for random upper semicontinuous functions

F m n

CHAPTER 2

SOME STRONG LAWS OF LARGE NUMBERS FOR DOUBLE ARRAYS OF RANDOM SETS WITH GAP TOPOLOGY

In this chapter, we establish some strong laws of large numbers for a double array of inde- pendent (or pairwise independent) random sets with the gap topology under various settings

2.1.The strong laws of large numbers for a double array of compactly uniformly integrable in the Ces`aro sense random sets

In this section, we introduce and prove a strong laws of large numbers for a double array ofpairwise independent, compactly uniformly integrable in the Ces`aro sense random sets

At first, we give some lemmas which will be used later

Lemma 2.1.1. Assume that {F mn : m ≥ 1, n ≥ 1} is a double array of compactly uniformly integrable in the Ces`aro sense random sets Then

(1) {f mn : m ≥ 1, n ≥ 1} is a double array of compactly uniformly integrable in the Ces`aro sense random elements, where f mn ∈ S0 , for all m ≥ 1, n ≥ 1.

(2) {s(x∗, F mn) : m ≥ 1, n ≥ 1} is a double array of uniformly integrable in the

Ces`aro sense random variables, for every x∗ ∈ S∗.

Lemma 2.1.2. Let A be a subset of X, B be the closed unit ball of a fixed arbitrary equivalent norm of X and let r > 0 If A is a compact set then

coA + r.B = co(A + r.B). (2.1)

The theorem below will establish the strong laws of large numbers for a double array ofpairwise independent, compactly uniformly integrable in the Ces`aro sense random sets withrespect to the gap topology based on Ter´an’s method and C.Castaing, N.V Quang, D.X.Giap’s technique.

Theorem 2.1.5 Let 1 ≤ p ≤ 2 and {X mn : m ≥ 1, n ≥ 1} be a double array of pairwise independent, compactly uniformly integrable in the Ces`aro sense random sets and satisfying the following conditions:

≤

Σ Σ

(2) there exists X ∈ k(X) such that

X s- lim inf (clE(X mn , X )), (2.2)

almost surely, in the gap topology, as m ∨ n → ∞.

2.2.The strong laws of large numbers for a double array of indepen- dent closed valued random variables in a

separable Banach space

In this section, we establish some strong laws of large numbers for double arrays of inde- pendent, pairwise independent closed valued random variables in a separable Banach space

At first, we introduce the strong laws of large numbers for a double array of random sets

in a Rademacher type p Banach space.

Theorem 2.2.2.Suppose that X is a Rademacher type p Banach space, where 1 ≤ p ≤ 2 Let

{X mn : m ≥ 1, n ≥ 1} be a double array of independent random sets satisfying the following conditions: