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

SOMELAWSOFLARGENUMBERSFOR MULTIDIMENSIONALARRAYSAND

NGHE AN -2 0 2 2

Workis completed at VinhU n i v e r s i t y

Supervisor: 1 Prof Dr Nguyen Van Quang

2 Dr Duong Xuan Giap

Reviewer1:

Reviewer2 :

Reviewer3 :

Thesiswillbedefendedatschool-levelthesis evaluating council at VinhUniversity

at , ,

Thesis canbefounda t :

- Vietnam NationalL ib rary

- Nguyen Thuc Hao Library and Information Center - VinhU n i v e r s i t y

PREFACE

1 Rationale

1.1.The limit theoremshavean important role in the development of

probabilitytheory.Theyhavebeen extensively studied and applied in several fields, such as

stochasticandintegralgeometry,mathematicaleconomics,statisticsandrelatedfields

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 canbeapplied in several fields such as optimization andcontrol, stochastic and integralgeometry,mathematical economics, etc.However,since the space

of closed subsets of Banach space does nothavethe structure of a vector space, there are severalirregularities in the study and establishment of limit theorems Therefore, the study ofnumericallawforrandomsetsisnotonlytheoreticalbutalsopractical

1.3 Formulti-indexed structure, the usual partial order relation is not complete So,

ifweextend the limit theorems for random sets from the sequence case to the multidimensionalarray case, thenwewillhavea lot of news thing This implies the results of multi-valued laws oflarge numbers morei n t e r e s t i n g

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.Forthe random sets case,

in the last 10 years, some interesting resultshavebeen introducedbyNguyenVanQuang, DuongXuan Giap, NguyenTranThuan, Hoang Thi Duyen, However,there are still many other resultsfor the single-valued random variables case thathavenot been extended to the random sets case

So, there willbemany interesting issues to study ifweextend the results valuedrandomvariablescasetoarraysofrandomsetscase

fromarraysofsingle-With the above reasons, we have chosen the topic for the thesis as follows:

4 Scope of theresearch

Thethesisfocusesonstudyingthelawsoflargenumbersfordoublearraysandtriangularly arrays ofrandom sets with gap topology Additionally, the thesis also establishes some laws oflargenumbersford-dimensionalarraysofrandomuppersemicontinuousfunctions

5 Methodology of theresearch

Weuse a combination of the fundamental methods of probabilitytheory, convexanalysisandfunctionalanalysissuchas:theconvexificationtechnique,theblockingprocedureinproving thelawoflargen u m b e r s

Inthisthesis,weextendtheP.Ter´an’sresultsfordoublearraysofrandomsetsandcom-biningP.Ter

´an’smethodwiththetechniquesforbuildingadoublearrayofselectionsdevelopedbyNguyenVanQuangandhisfellowstoprovethe“liminf”partofPainlev´e-Kuratowskicon-vergence Using theseresults,weestablish some laws of large numbers for double arrays of random sets with the gaptopology for the casem∨n→ ∞.

Forthe triangular arrays of random sets,weestablish some strong laws of large numbers ofrowwise independent random sets, compactly uniformly integrable and satisfying some variousconditions.Todo this,wepresent some strong laws of large numbers for triangular arraysofrandomelementsandprovethe“liminf”partofPainlev´e-

Kuratowskiconvergence.Finally,byextending the results that NguyenVanQuang and DuongXuan Giap introduced in 2013,weestablish the stronglawof large numbers for triangular arrays

of random sets taking closed-valuesof Rademacher typepBanach space.

Extending some results that NguyenTranThuan and NguyenVanQuang introduced in2016forthecasemultidimensionalarrays,weobtainsomelawsoflargenumbersfornegatively

associatedandpairwisenegativelydependentrandomuppersemicontinuousfunctions.Toprovetheselimittheorems,wealsointroduceH´ajek-R´enyi’stypemaximalinequalityforanarrayofnegativelyassociated random upper semicontinuous functions and thelawof large numbers ford-

dimensional arrays of pairwise negatively dependent real-valued randomvariables

7.2 The organization of ther e s e a r c h

Besidesthesectionsofusualnotations,preface,generalconclusions,andrecommendations, list of theauthor’s articles related to the thesis and references, the thesis is organized into three chaptersanda p p e n d i x

Chapter 1 introduces somep r e l i m i n a r i e s

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.

AppendixprovidessomemaximalinequalitieswhichformRosenthal’stypeandHa´jek-R

´enyi’stypeformulti-dimensionalstructureandestablishessomelawsoflargenumbersfor

d-pendentrandomuppersemicontinuousfunctionsundervarioussettings

1.1.The convergence on the space of closed subsets of

are lower limit and upper limit of the array{An :n∈Nd}, relative to topologytas|n| → ∞.

Definition1.1.6 LetA∈ c(X) The array{An :n∈Nd}⊂c(X) is said tobe

(1)convergesin the sense ofMoscotoAas|n| → ∞and is denotedbyAn (M)

AsliceofaballistheintersectionofaclosedballB¯(x,r)(wherex∈X,r>0)andofaclosed half

spaceF(x∗,α) ={x∈X:⟨x∗,x ≥⟩ ≥ α}(wherex∗∈X∗,x∗̸=0 andα∈R)

(S)

(4)convergesin theslice topologytoAkhi|n|→ ∞and is denotedbyAn −→Aas

|n| → ∞or

(S)-|lim An =A,if lim

|n|→∞D(An, C) =D(A, C) for all nonempty slicesCof balls

In this part, we introduce the random sets and some basic notions

LetBc(X)be theσ-field onc(X) generated by the sets

U−:={C∈c(X) :C∩U̸= }∅} ,

forallopensubsetsUo fX.WecallBc(X)theEffr¨osσ-field.

Definition 1.2.1.A mappingX: Ω→c(X) is said to beF-measurable if for everyB∈

Bc(X),X−1(B)∈F.The mappingF-measurableXis also calledF-measurable random set.

IfF=AthenXis said for shortly to berandom set.

LetXbea random set,wedefineA X ={X−1(B),B B∈ c(X)}.T h e n AX is the smallest

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

f: Ω→Xsuch that the norm∥f∥ p

=(E∥f∥ p )p1is finite If F=AthenLp(F, X) is denotedfor shortlybyLp(X) IfX=RthenLp replaceL p(R)

For each random setF-measurableXand forp≥1, we denoteS p (F)={f∈L p(F,X) :

f(ω)∈X(ω)a.s.} In the caseF=AthenS p (A) is replacedbyS p

Definition1.2.4 TherandomsetX:Ω→c(X)iscalledintegrableifS1

is nonempty

Definition1.2.5. Theexpectationo f integr abler ando m s et X,d e n o t e d byEX,i s de f i ne d

by

EX:={Ef:f∈S1}whereEfis the usual Bochner integral of random elementf.

In Definition 1.1.6, ifwereplaceAn byXn(ω)and replaceAbyX(ω)forωin a set with probability

1, whereX,Xn,n∈Ndare random sets, thenweobtain the definition of almost sure convergence forrandomsets

1.3.The uniformly integrable compactness and the uniformly boundedofanarrayofrandomsets

Wepresent some notions: compactly uniformly integrable, compactly uniformlyintegrableintheCes`arosense,uniformlyboundedfordoublearrayandtriangulararrayofrandomele-

ments and random sets

Definition1.3.1 (1) A double array of random elements{f mn :m≥1, n≥1}is said tobe

LetAbe a subset of Ω, the complement of the setAwith respect to Ω is denoted byA c.(3) A double array of random sets{X mn :m≥1, n≥1}is said

supE.∥X ni∥1(X ni(.)⊂K) cΣ <ε.

n,

i

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

boundedbya random variableξif for alln≥1, 1≤i≤nand for every real numbert >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

variableξif for alln≥1, 1≤i≤nand for every real numbert >0,

P(∥X ni∥> t)≤P(|ξ|> t).

1.4.The random upper semicontinuousfunctions

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

LetKbethe set of compactintervalsofR Ifxis an element ofK, then it

willbedenotedbyx=[x(1);x(2)], wherex(1), x(2)aretwoend points The Hausdorff distanced H onKis

definedby

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

Fora functionu:R→[0; 1], theα-level setofuis definedby[u] α ={x∈R:u(x)≥α}for eachα∈(0;

1].Foreachα∈[0; 1), [u] α+denotes the closure of{x∈R:u(x)>α} In particular, [u]0+iscalled thesupportofuand denotedbysuppu The level set [u] α ofuis closed for allα∈(0; 1] iffuis

upper semicontinuous function A upper semicontinuous functionu:R→[0; 1] iscalledquasiconcave functionifu(λxx+(1−λx)y)≥min{u(x), u(y)}for allx, y∈R, anditsequivalentcondition isthat[u] α is aconvexsubset ofRfor everyα∈(0; 1] LetUdenote the family

of all upper semicontinuous functionsu:R→[0; 1] satisfying the followingc o n d i t i o n s

(1) suppuis compact;

(2) [u]1̸=∅};

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

saidtobenegativelyassociatedifforanydisjointsubsetsA1,A2of{1,2, ,n}andanyrealcoordinatewi

senondecreasingfunctionsfonR|A1 |,gonR|A2 |suchthat

(2) Let{Xn,n∈Nd}bean array ofK-valued random variables Then,{Xn,n∈Nd}is saidtobenegativelyassociatedif{X(1),n∈Nd}and{X(2),n∈Nd}are arrays of negatively associatedreal-valued random variables

(3) Let{Xn,n∈Nd}bean array ofU-valued random variables Then{Xn,n∈Nd}issaidtobelevel-wisenegativelyassociatedif{[Xn]α ,n∈Nd}arearraysofnegativelyassociatedK-valued

random variables for allα∈(0;1]

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

Y

n n

saidtobenegativelydependentifthetwofollowinginequalitieshold

n

P(X1>x1, ,X n >x n)≤ P(X i >x i),

i=1 n

P(X1≤x1, ,X n ≤ x n)≤ P(X i ≤ x i),

i=1

for allx i ∈R,i=1,2, , n.

Aninfinitecollectionofreal-valuedrandomvariablesisnegativelydependentifeveryfinite subfamily isnegativelyd e p e n d e n t

The conclusions of Chapter 1

In this chapter, we obtain some main results:

- Introduces o m e i m p o r t a n t c o n v e r g e n c e s o n t h e s p a c e o f c l o s e d s u b s e t s o f B a n a c h s

p a c e ;

- Presentrandomsetsandsomeandsomerelatedconcepts;

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

F m n

CHAPTER 2

SOME STRONGLAWSOF LARGE NUMBERSFORDOUBLEARRAYSOF 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.Thestronglawsoflargenumbersforadoublearrayofcompactlyunifo rmlyintegrableintheCes`arosenserandomsets

In this section,weintroduce andprovea strong laws of large numbers for a double

arrayofpairwiseindependent,compactlyuniformlyintegrableintheCes`arosenserandomsets

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

Lemma2.1.1.

Assumethat{F mn :m≥1,n≥1}isadoublearrayofcompactlyuniformlyintegrableintheCes`arosens erandomsets.T h e n

(1) {f mn :m≥1,n≥1}isadoublearrayofcompactlyuniformlyintegrableintheCes`aro

senserandomelements,wheref mn ∈S0 , forallm≥1, n≥1.

(2) {s(x∗,F mn):m≥1,n≥1}isadoublearrayofuniformlyintegrableintheCes`arosense random variables, for every x∗∈S∗.

Lemma2.1.2. LetAbeasubsetof X ,Bbetheclosedunitballofafixedarbitraryequivalentnorm ofXand let r >0 If A is acompactsetthen

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

∥ ∥

≤

ΣΣΣ

The theorem below will establish the strong laws of large numbers for a doublearrayofpairwiseindependent,compactlyuniformlyintegrableintheCes`arosenserandomsetswithrespecttothegaptopologybasedonTer

´an’smethodandC.Castaing,N.V.Quang,D.X.Giap’stechnique

Theorem 2.1.5.Let1≤p≤2and{X mn :m≥1, n≥1}bea doublearrayof pairwiseindependent,compactlyuniformlyintegrableintheCes`arosenserandomsetsandsatisfying thefollowing conditions:

2.2.Thestronglawsoflargenumbersforadoublearrayofindepen-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.Atfirst,weintroduce the strong laws of large numbers for a double array of random sets in

a Rademacher typepBanachspace.

Theorem2.2.2. Supposethat X isaRademachertypepBanachspace,where1≤p≤2.Let

{X mn :m≥1,n≥1}beadoublearrayofindependentrandomsetssatisfyingthefollowingconditions:

The following theorem will establish the strong laws of large numbers for a double array ofpairwiseindependentidenticallydistributedrandomsetswithrespecttothegaptopology.

Theorem2.2.4.Supposethat{F mn :m≥1,n≥1}isadoublearrayofpairwiseindependentrandomsets,h avingthesamedistributionasXwhichisak(X)-valuedrandomsetsatisfyingE(∥X∥log+∥X∥)<∞.Then,

1

F

mn i=1j=1 ij(ω)→coEX almost surely, in the gap topology, as m∨n→ ∞.

The conclusions of Chapter 2

In this chapter, we obtain some main results:

- Establishsomestronglawsoflargenumbersfordoublearraysofrandomsetsinthecases:pairewiseindependentandcompactlyuniformlyintegrableintheCes`arosense,orpairewiseindependent

identicallyd i s t r i b u t e d

- Provethe strong laws of large numbers for double arrays of independent random sets in aRademacher typepBanachs p a c e

RANDOM SETS WITH GAP TOPOLOGY

In this chapter, at first,weprovesome strong laws of large numbers of triangular arrays ofrandom elements Nextweestablish some strong laws of large numbers of triangular arraysofrandomsetswithrespecttothegaptopology

3.1.Thestronglawsoflargenumbersoftriangulararraysofrandom elements

In this section, we will establish some strong laws of large numbers of triangular arrays ofrandom elements under various settings This results will be used to prove some strong laws oflarge numbers for triangular arrays of random sets

Theorem3.1.1.Let{f ni :n≥1,1≤i≤n}beatriangulararrayofrowwiseindependentandcompac tlyuniformlyintegrablerandom elements in aseparableBanachspace Letψ(t)bea positive, even, convex, continuous function suchthat

< ;

ψ(n)

n=1i=1

Σ 1Σn Σ

3.2.Thestronglawsoflargenumbersoftriangulararraysofrandom sets

In this section, the firsttwotheorems will establish some strong laws of large numbers fortriangular arrays of rowwise independent and compactly uniformly integrable random sets withrespect to the gapt o p o l o g y

Theorem3.2.2.Let{X ni :n≥1,1≤i≤n}beatriangulararrayofrowwiseindependentandcompactl yuniformlyintegrablerandomsetsinaseparableBanachspace.Letψ(t)beapos-

itive,even,convex,continuousfunctionandψ(t)satisfies(3.1)and(3.2)forsomenonnegativeintegerr.If the following conditionsaresatisfied