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.

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

Supervisor: 1 Prof Dr Nguyen Van Quang

2 Dr Duong Xuan Giap

Thesis can be found at:

- Vietnam National Library

- Nguyen Thuc Hao Library and Information Center - Vinh University

1 Rationale

1.1 The limit theorems have an important role in the development of probability theory Theyhave 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 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

proba-1.3 For multi-indexed structure, the usual partial order relation is not complete So, if weextend the limit theorems for random sets from the sequence case to the multidimensional arraycase, then we will have a lot of news thing This implies the results of multi-valued laws oflarge numbers more interesting

1.4 In Vietnam, the limit theorems for single-valued random variables vector space, there areseveral 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

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”

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 and triangularlyarrays of random sets with gap topology Additionally, the thesis also establishes some laws oflarge numbers for d-dimensional arrays of random upper semicontinuous functions

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 in provingthe 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 selections developed

com-by Nguyen Van Quang and his fellows to prove the “liminf” part of Painlev´e-Kuratowski vergence Using these results, we establish some laws of large numbers for double arrays ofrandom sets with the gap topology for the case m ∨ n → ∞

con-For the triangular arrays of random sets, we establish some strong laws of large numbers ofrowwise independent random sets, compactly uniformly integrable and satisfying some variousconditions To do this, we present some strong laws of large numbers for triangular arrays ofrandom elements and prove the “liminf” part of Painlev´e-Kuratowski convergence Finally, byextending the results that Nguyen Van Quang and Duong Xuan Giap introduced in 2013, weestablish 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 negativelyassociated and pairwise negatively dependent random upper semicontinuous functions To provethese limit theorems, we also introduce H´ajek-R´enyi’s type maximal inequality for an array ofnegatively associated random upper semicontinuous functions and the law of large numbers ford-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, and recommendations,list of the author’s articles related to the thesis and references, the thesis is organized into threechapters 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

independent random sets with gap topology.

Appendix provides some maximal inequalities which form Rosenthal’s type and H´R´enyi’s type for multi-dimensional structure and establishes some laws of large numbers ford-dimensional arrays of level-wise negatively associated and level-wise pairwise negatively de-pendent random upper semicontinuous functions under various settings

ajek-CHAPTER 1

PRELIMINARIES

In this chapter, we introduce some important types of convergence on the space of closedsubsets of Banach space, some preliminaries of random sets and present some basic notions andrelated 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 closedsubsets, nonempty c(X) of X We put

where {Ank : k ∈Nd} is a sub-array of {An: n ∈Nd} The sets t- lim inf

|n|→∞An and t- lim sup

|n|→∞

Anare 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

(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 −−→ A as |n| → ∞ or (K)- lim(K)

|n|→∞An = A, ifs- lim sup

|n|→∞

An = s- lim inf

|n|→∞An = A

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

|n|→∞An = A if and only if w- lim sup

|n|→∞

An ⊂ A ⊂ s- lim inf

|n|→∞An.(2) On the space of convex compact and non-empty subsets, Wijsman convergence leads toPainlev´e - Kuratowski convergence On the space of convex, closed and non-empty subsets, thetypes 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 Effr¨os σ-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 = Athen X is said for shortly to be random set

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

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

on Bc(X) defined by

PX(B) = P X−1(B)
, B ∈ Bc(X)

Definition 1.2.2 A family of random sets {Xi, i ∈ I} is said to be independent (respectively,pairwise independent ) if {AXi, i ∈ I} are independentl (respectively, pairwise independent), and

is said to be identically distributed if all PXi, i ∈ I are identical

Definition 1.2.3 A random element f : Ω →X is called a selection of the random set X if

f (ω) ∈ X(ω) a.s

For 1 ≤ p < ∞, Lp(Ω, A, P,X) = Lp(X) denotes the space of A-measurable functions

f : Ω →X such that the norm ∥f ∥p = (E∥f ∥p)p1 is finite If F = A then Lp(F ,X) is denotedfor shortly by Lp(X) If X=R then Lp replace Lp(R)

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

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

Definition 1.2.4 The random set X : Ω → c(X) is called integrable if SX1 is nonempty.Definition 1.2.5 The expectation of integrable random set X, denoted by EX, is definedby

EX := {Ef : f ∈ SX1}

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 withprobability 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 {fmn: m ≥ 1, n ≥ 1} is said to beuniformly integrable in the Ces`aro sense if

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

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

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

Definition 1.3.3 (1) A triangular array of random elements {fni : n ≥ 1, 1 ≤ i ≤ n} isuniformly bounded by a random variable ξ if for all n ≥ 1, 1 ≤ i ≤ n and for every real number

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 ofR 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 dH on K isdefined by

dH(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 semicontinuousfunction u : R → [0; 1] is called quasiconcave function if u(λx + (1 − λ)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 ̸= ∅;

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

Definition 1.4.9 (1) A finite family {Xi, 1 ≤ i ≤ n} of real-valued random variables issaid to be negatively associated if for any disjoint subsets A1, A2 of {1, 2, , n} and any realcoordinatewise nondecreasing functions f onR|A1 |, g on R|A2 | such that

Covf1(Xi, i ∈ A1), f2(Xj, j ∈ A2)≤ 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 {Xn(1), n ∈ Nd} and {Xn(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 negatively associatedK-valued random variables for all α ∈ (0; 1]

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

said to be negatively dependent if the two following inequalities hold

(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 {Xn(1), n ∈ Nd}and {Xn(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 negatively dependent)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 semicontinuousfunctions

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 pendent (or pairwise independent) random sets with the gap topology under various settings

inde-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

of pairwise 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 {Fmn : m ≥ 1, n ≥ 1} is a double array of compactly uniformlyintegrable in the Ces`aro sense random sets Then

(1) {fmn: m ≥ 1, n ≥ 1} is a double array of compactly uniformly integrable in the Ces`arosense random elements, where fmn ∈ S0

The theorem below will establish the strong laws of large numbers for a double array

of pairwise 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’stechnique

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

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

2.2 The strong laws of large numbers for a double array of dent closed valued random variables in a separable Banach space

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

inde-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 thatXis a Rademacher type p Banach space, where 1 ≤ p ≤ 2 Let{Xmn : m ≥ 1, n ≥ 1} be a double array of independent random sets satisfying the followingconditions: