Sự hội tụ của tổng các phần tử ngẫu nhiên phụ thuộc nhận giá trị trong không gian hilbert tt tiếng anh

Objective of the research In this thesis, we establish the strong law of large numbers, the weak law of large numbersand complete convergence for sequences of dependent random elements i

VINH UNIVERSITY

*

-NGUYEN THI THANH HIEN

CONVERGENCE FOR SUMS

OF DEPENDENT RANDOM ELEMENTS IN

Supervisor: Assoc Prof Dr Le Van Thanh

Reviewer 1:

Reviewer 2:

Reviewer 3:

The thesis will be defended at the university-level thesis

evaluating council at Vinh University

at , ,

The thesis can be found at:

- Vietnam National Library

- Nguyen Thuc Hao Library and Information Center - Vinh University

1 Rationale

1.1 The law of large numbers is a classical problem of probability theory, it asserts that theadditive mean of independent random variables distributions converges in a certain sense onthe expectation of random variables However, large numbers of mathematicians continue to

be interested in studying mathematics and have many applications in statistics, econometrics,natural sciences and many other fields Therefore, the study of numerical law is not onlytheoretical but also practical

1.2 When we study about probability theorem, the independence of random variables isimportant However, random phenomena that occur in practice often depend on each other

So, we have to study different types of dependencies of random variables to fit practical such

as local dependence, negative association, negative dependence

1.3 The development of limit theorems in probability theory has led to more general resultsthan classical results One of the general directions is that of the results obtained for thereal-valued random variables that are given to the values in the Hilbert space When studyingabout limit theorems, many authors have obtained interesting results, such as Gilles Pisier,Michel Talagrand, Andrew Rosalsky, Pedro Teran, Nguyen Van Quang

1.4 The convergence of weighted sums of random variables has many important applications instochastic control and mathematical statistics , these are the nonparametric multiple regressionmodel and the least squares estimators

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

“ Convergence for sums of dependent random elements in Hilbert

spaces ”

2 Objective of the research

In this thesis, we establish the strong law of large numbers, the weak law of large numbersand complete convergence for sequences of dependent random elements in Hilbert spaces

3 Subject of the research

The research object of the thesis is:

- Negatively associated random elements, negative dependence and pairwise negative

de-pendence for random elements in Hilbert spaces;

- Some limit theorems

4 Scope of the research

The thesis focuses on the study the dependence in probability theory, the convergence forsums of random variables

5 Methodology of the research

- We use the independent method of document study and we analyze the existing results,then develop them into models with similar structures, or more general models

- We establish seminar groups under the guidance of instructors, and exchange with localand foreign scientists

6 Contributions of the thesis

The results of the thesis contribute to enriching the research direction of the law of largenumbers and complete convergence for sequences of dependent random elements in Hilbertspaces

The thesis can be used as a reference for students, masters students and PhD studentsbelong to the specialty: Theory of Probability and Mathematical Statistics

7 Overview and organization of the research

7.1 Overview of the research

The concepts of pairwise negative dependence, negative dependence and negative ation were introduced by Lehmann, by Ebrahimi and Ghosh and by Joag-Dev and Prochan.Joag-Dev and Proschan pointed out that many useful distributions enjoy the negative asso-ciation properties (and therefore they are negatively dependent) including multinomial dis-tribution, multivariate hypergeometric distribution, Dirichlet distribution, and distribution ofrandom sampling without replacement Because of its wide applications in multivariate statisti-cal analysis and reliability, the notion of negative association has received considerable attentionrecently We refer the reader to Joag-Dev and Prochan for fundamental properties, Newmanfor the central limit theorem, Matula for the three series theorem, Shao for the Rosenthal typemaximal inequality and the Kolmogorov exponential inequality

associ-The concept of association, a dependence structure stronger than pairwise negative dence, was first extended to Hilbert space valued random elements by Burton, Dabrowski and

depen-Dehling in 1986 Ko, Kim and Han introduced the concept of negative association for randomelements with values in real separable Hilbert spaces In 2014, Huan, Quang and Thuan presentanother concept of coordinatewise negative association for H-valued random elements which ismore general than the concept of Ko, Kim and Han In many papers one can find some inter-esting results concerning sequences of H-valued negatively associated radom variables We referonly some of them Almost sure convergence by Ko, Kim and Han; almost sure convergenceextending the results of Ko, Kim and Han by Thanh; the Hajek-Renyi inequality by Miao; theBaum-Katz type theorem by Huan, Quang and Thuan

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

Chapter 1 presents some preliminaries

Chapter 2 studies some limit theorems such as laws of large numbers and complete gence for sequences of pairwise and coordinatewise negative dependence for random elements

conver-in Hilbert spaces

Chapter 3 studies some limit theorems such as laws of large numbers and complete vergence for sequences of coordinatewise negatively associated for random elements in Hilbertspaces

coordinate-1.1 The negatively dependent random variables, the negatively ciated random variables

asso-1.1.1 Definition A collection of random variables {X1, X2, , Xn} is said to be

i) negatively lower dependent, if for all x1, , xn ∈R, we have

P(X1≤ x1, , Xn ≤ xn) ≤P(X1≤ x1) P(Xn ≤ xn), (1.1)ii) negatively upper dependent, if for all x1, , xn ∈R, we have

P(X1 > x1, , Xn > xn) ≤P(X1> x1) P(Xn > xn), (1.2)iii) negatively dependent, if they are both negatively lower dependent and negatively upperdependent

A sequence of random variables {Xi, i ≥ 1} is said to be negatively dependent if for any

n ≥ 1, the collection {X1, , Xn} is negatively dependent

1.1.2 Definition A collection of random variables {X1, , Xn} is said to be negatively ciated if for any disjoint subsets A, B of {1, , n} and any real coordinatewise nondecreasing

asso-functions f on R|A| and g on R|B|,

Cov(f (Xk, k ∈ A), g(Xk, k ∈ B)) ≤ 0 (1.3)whenever the covariance exists, where |A| denotes the cardinality of A

A sequence {Xn, n ≥ 1} of random variables is said to be negatively associated if everyfinite subfamily is negatively associated

1.2 The negatively associated random elements, the negatively pendent random elements

de-1.2.1 Definition Let H be a real separable Hilbert space with orthonormal basis {ej, j ∈ B}and inner product h·, ·i A sequence {Xi, i ≥ 1} of random elements with values in H is said

to be negatively associated (NA) if for any d ≥ 1, the sequence of Rd-valued random elements{(hXi, e1i, , hXi, edi), i ≥ 1} is negatively associated

1.2.2 Definition A sequence {Xi, i ≥ 1} of random elements taking values in H is said to becoordinatewise negatively associated (CNA) if for some orthonormal basis {ej, j ≥ 1} and foreach j ≥ 1, the sequence of random variables {hXi, eji, i ≥ 1} is negatively associated

In this section, we introduce the notion of coordinatewise negative dependence (CND), and wise and coordinatewise negative dependence (PCND) for random elements in Hilbert spaces.1.2.3 Definition A sequence {Xi, i ≥ 1} of random elements taking values in H is said to

pair-be coordinatewise negative dependence (CND) (resp., pairwise and coordinatewise negative pendence (PCND)) if for some orthonormal basis {ej, j ≥ 1} and for each j ≥ 1, the sequence

de-of random variables {hXi, eji, i ≥ 1} is negatively dependent (resp., pairwise negatively dent)

depen-The next, we prove the Rademacher-Menshov type inequality and the H´ajek-R´enyi type equality for sums of PCND random elements in H

in-1.2.4 Theorem Let {Xn, n ≥ 1} be a sequence of PCND mean 0 random elements in Hsatisfying EkXnk2< ∞ for all n ≥ 1 Then for any n ≥ 1, we have

2

≤ 2

Pm i=1EkXik2

b2 m

1.3 The slowly varying function

1.3.1 Definition A real-valued function R(·) is said to be regularly varying function withindex of regular variation ρ (ρ ∈ R) if it is a positive and measurable function on [A, ∞) forsome A > 0, and for each λ > 0,

lim

x→∞

R(λx)R(x) = λ

(i) There exists B ≥ A such that xpL(x) is increasing on [B, ∞), x−pL(x) is decreasing

on [B, ∞) and limx→∞xpL(x) = ∞, limx→∞x−pL(x) = 0

(iii) For all λ > 0,

lim

x→∞

L(x)L(x + λ) = 1.

Because of Lemma 1.3.2, we have

1.3.3 Lemma Let p > 1, q ∈ R, and let L(·) be a differentiable slowly varying functiondefined on [A, ∞) for some A > 0 satisfying

lim

x→∞

xL0(x)L(x) = 0, Then for n large enough, we have

2Lq(n)3(p − 1)np−1 ≤

The following proposition gives a criterion for E (|X|αLα(|X| + A)) < ∞

1.3.4 Proposition Let α ≥ 1 and let X be a random variable Let L(·) be a slowly varyingfunction defined on [A, ∞), for some A > 0 Assume that xαLα(x) and x1/αL(xe 1/α) areincreasing on [A, ∞) Then

E(|X|αLα(|X| + A)) < ∞ if and only if

X

n≥A α

P(|X| > bn) < ∞ (1.10)

where bn = n1/αL(ne 1/α), n ≥ Aα

The conclusions of Chapter 1

In this chapter, we obtain some main results:

- A brief some of the basic concepts and properties of negatively dependent randomvariables, pairwise negatively dependent random variables, negatively associated randomvariables;

- A brief some of the basic concepts and properties of negatively associated randomelements, coordinatewise negatively associated random elements in Hilbert spaces;

- Present the new notion of coordinatewise negative dependence (CND), and pairwise andcoordinatewise negative dependence (PCND) for random elements in Hilbert spaces;

- Present and prove some classical inequalities of random elements in Hilbert spaces;

- A brief some of the basic concepts and properties of regularly varying function, slowlyvarying function Present and prove some properties of regularly varying function, slowlyvarying function

CHAPTER 2

THE LAW OF LARGE NUMBERS AND COMPLETE

CONVERGENCE FOR SEQUENCES OF PAIRWISE AND COORDINATEWISE NEGATIVE DEPENDENCE FOR

RANDOM ELEMENTS IN HILBERT SPACES

In this chapter, we establish some limit theorems kind of law of large numbers and completeconvergence for sequences of pairwise and coordinatewise negative dependence for randomelements in Hilbert spaces

2.1 The strong law of large numbers and complete convergence

In this section, we establish some strong laws of large numbers and complete convergence forsequences of pairwise and coordinatewise negative dependence for random elements in Hilbertspaces

The following theorem is an extension of the classical the Rademacher-Menshov strong law

of large numbers to the blockwise PCND random elements in Hilbert spaces

2.1.1 Theorem Let {Xn, n ≥ 1} be a sequence of mean 0 random elements in H such thatfor any k ≥ 0, the random elements {Xi, 2k ≤ i < 2k+1} are PCND Let {bn, n ≥ 1} be anondecreasing sequence of positive constants satisfying

The next theorem in this section establishes the complete convergence for sequences of PCNDelements in Hilbert spaces Let X be a random vector in H Here and thereafter, we denotethe jth coordinate of X by X(j), i.e.,

In the special case, αp = 1 and ani ≡ 1, we have a following corollary

2.1.3 Corollary Let {Xn, n ≥ 1} be a sequence of PCND identically distributed mean 0 dom elements in H, and let 1 ≤ p < 2 If

Pn i=1Xi(n log2n)1/p → 0 a.s as n → ∞

2.2 The weak law of large numbers

In the following theorem, we establish the weak law of large number for sequences of PCNDelements in Hilbert spaces

2.2.1 Theorem Let {Xn, n ≥ 1} be a sequence of PCND random elements in H and {bn, n ≥1} be a sequence of positive constants For n ≥ 1, k ≥ 1, j ∈ B, we set

In Chapter 2 of the thesis, some limit theorems such as laws of large numbers and the Baum

- Katz type theorem on complete convergence for sequences of pairwise and coordinatewisenegative dependent random elements are established To prove strong laws of large numbers

and the Baum - Katz type theorem, we often have to use the maximal type inequalities as avery important tool However, for the pairwise and coordinatewise negative dependent randomelements, the Kolmogorov type inequality does not hold any more Instead, we have to provethe Rademacher-Menshov type inequality for this dependence struture To prove results onlaw of large numbers, we estimate the partial sums of random elements in each block [2k, 2k+1)and then apply a sub-sequences method This approach allows us to consider a sequence ofblockwise, pairwise and coordinatewise negative dependent random elements For the weak laws

of large numbers, we generalized the Feller weak law by considering the normalizing constants

bn = nαL(n), where L(n) is a slowly varying function To achieve this result, we need to provesome properties of the slowly varying function, as shown in Lemma 1.3.6, Lemma 1.3.9 andProposition 1.3.10 Besides, we also present some examples to illustrate the sharpness of themain theorems (Example 2.1.9, 2.2.2 and 2.2.3)

CHAPTER 3

THE LAW OF LARGE NUMBERS AND COMPLETE

CONVERGENCE FOR SEQUENCES OF COORDINATEWISE NEGATIVELY ASSOCIATED RANDOM ELEMENTS IN

HILBERT SPACES

In this chapter, we establish some limit theorems kind of law of large numbers and completeconvergence for sequences of coordinatewise negatively associated random elements in Hilbertspaces

3.1 The strong law of large numbers and complete convergence

In the following theorem, we establish complete convergence for weighted sums of natewise negatively associated and identically distributed random elements in Hilbert spaces.3.1.1 Theorem Let 1 ≤ p < 2, αp ≥ 1, {X, Xn, n ≥ 1} be a sequence of coordinatewisenegatively associated and identically distributed random elements in H and let L(x) be a slowlyvarying function defined on [A, ∞) for some A > 0 When p = 1, we assume further thatL(x) ≥ 1 and is increasing on [A, ∞) Let bn = nαL(ne α), n ≥ A1/α and {ani, n ≥ 1, 1 ≤ i ≤ n}are constants satisfying

3.1.2 Corollary Let 1 ≤ p < 2, {X, Xn, n ≥ 1} be a sequence of coordinatewise negativelyassociated and identically distributed random elements in H and let L(x) be a slowly varyingfunction defined on [A, ∞) for some A > 0 When p = 1, we assume further that L(x) ≥ 1 and

is increasing on [A, ∞) Let bn = n1/pL(ne 1/p), n ≥ Ap If random element X satisfies

is increasing on [A, ∞) Let bn = n1/pL(ne 1/p), n ≥ Ap If random element X satisfies

However, H is a finite dimensional Hilbert space We have the following theorem

3.1.4 Theorem Let 1 ≤ p < 2, αp ≥ 1 and {e1, e2, , ed} be an orthonormal basis in H.{X, Xn, n ≥ 1} be a sequence of coordinatewise negatively associated and identically distributedrandom elements in H and let L(x) be a slowly varying function defined on [A, ∞) for some

A > 0 When p = 1, we assume further that L(x) ≥ 1 and is increasing on [A, ∞) Let

bn = nαL(ne α), n ≥ A1/α Then the following four statements are equivalent

i) The random element X satisfies

E(X) = 0, E (kXkpLp(kXk + A)) < ∞ (3.8)ii) For every array of constants {ani, n ≥ 1, 1 ≤ i ≤ n} satisfying

n

X

i=1