Weak laws of large numbers in banach spaces

2 Weak laws of large numbers for Banach space-valued random2.1 Denition and some well-known classical results.. 102.2 Weak laws of large numbers in Banach spaces with Schauder basis 132.

THE UNIVERSITY OF DANANG

UNIVERSITY OF EDUCATION

FACULTY OF MATHEMATICS

DUONG PHUOC LUAN

WEAK LAWS OF LARGE NUMBERS

IN BANACH SPACES

UNDERGRADUATE THESIS

April, 2016

THE UNIVERSITY OF DANANG

UNIVERSITY OF EDUCATION

FACULTY OF MATHEMATICS

DUONG PHUOC LUAN

WEAK LAWS OF LARGE NUMBERS

First of all, I would like to express my deepest gratitude to my supervisor,

Dr Cao Van Nuoi, for his excellent guidance, caring, patience, and providing mewith an excellent atmosphere for writing this thesis I have learned a lot fromhim not only in scientic environment but also in everyday life

I would like to thank all the teachers who have taught me from the rst day

of my university life till now I am also grateful to the leader boards of Faculty

of Mathematics, University of Education - Danang University for permitting me

to write this thesis in English

Last but not least, I also thank my close friends who have supported me toovercome tough times I greatly value their friendship and I deeply appreciatetheir belief in me

Danang, 25 April 2016

Duong Phuoc Luan

2 Weak laws of large numbers for Banach space-valued random

2.1 Denition and some well-known classical results 102.2 Weak laws of large numbers in Banach spaces with Schauder basis 132.3 Weak laws of large numbers in Rademacher type p Banach spaces 20

3 Weak laws of large numbers for Banach space-valued

3.1 Some properties of Banach space valued martingales 243.2 Weak laws of large numbers for Banach space-valued martingales 27

In probability, the limit theorems in general and the laws of large numbers

in particular have been studying by many mathematicians The laws of largenumbers have many applications in statistics, economics, medicine and otherempirical sciences Therefore, the study of it has not only the theoretical mean-ings but also the empirical meanings

The classical laws of large numbers are mainly for random variables takingreal values The great amount of interest over the last 60 years in representingstochastic processes as random elements in a Banach space has inspired the study

of the laws of large numbers for random elements taking their values in a Banachspace The results obtained in this study have the rigid relationship with thegeometry of Banach spaces and form the interference between probability theoryand functional analysis

This thesis presents some results about the weaks laws of large numbers inBanach spaces

The thesis is organised as follow:

Chapter 1 presents the basic knowledge about functional analysis and probability

on Banach spaces

Chapter 2 presents the weak laws of large numbers for random elements takingtheir values in a Banach space Firstly, we present the denition of the weaklaw of large numbers and some well-known results in the real case Secondly, wemention the weaks laws of large numbers in Banach spaces with Schauder basisand the extension of those results for arbitrary real separable Banach space byembedding each space isomorphically in the Banach space C[0; 1] Thirdly, wemention a weak law of large numbers for random elements taking their values in

a Rademacher type p Banach space (1≤ p ≤2)

Chapter 3 presents a weak law of large numbers for martingales in a Banachspace Some properties of martingales in Banach space are also examined

CHAPTER 1

PRELIMINARIES

1.1 Banach spaces

Denition 1.1.1 LetEbe a vector space A norm onE is a real valued function

k.k on E such that the following conditions are satised by all element x and y

in E and each scalar α:

(i) kxk ≥0, and kxk= 0 if and only if x= 0E;

(ii) kαxk=|α|kxk;

(iii) kx+yk ≤ kxk+kyk (triangle inequality)

The pair (E, kk)is called a normed space or normed vector space or normed linearspace

If E is a real vector space, then E is called a real normed space

Let d be a function from E2 into R dened by

com-If E is a real vector space and (E, k.k) is a Banach space, then (E, k.k) is called

a real Banach space

kf k= sup

kxk≤1

f(x) .

Theorem 1.1.3 (Hahn-Banach, [7]) LetE be a normed space andF is a subspace

of E Then, for each f ∈ F∗, there exists ˆf ∈ E∗ such that fˆ = kf k and therestriction of ˆf to F is f

Corollary 1.1.4 ([7]) If xis a nonzero element of a normed space E, then thereexists an f ∈ E∗ such that kf k= 1 and f(x) =kxk

Corollary 1.1.5 ([7]) If x and y are dierent elements of a normed space E,then there exists an f ∈ E∗ such that f(x)6=f(y)

Denition 1.1.6 A normed space E is said to be separable if it has a countabledense subset, that is, there exists a sequence (xn) ⊂ E such that if x ∈ E and

ε >0are arbitrary, then kx − xnk < ε for some n ∈N.

Denition 1.1.7 Let (E, k.k) be a normed space A sequence (x n) ⊂ E is aSchauder basis for E if, for each x ∈ E, there exists a unique sequence of scalars(an) such that

x=

∞Xn=1

anxn.

A Schauder basis (xn) Pn

k=1 akxk

n≥1

is monotone increasing for each sequence of scalars (an)

When a norm space E has a Schauder basis (x n), we dene the coordinatefunctionals x∗n: E →R by

x∗n(x) =a n ,

where x =

∞X

n=1

anxn It is straightforward that each x∗n is linear and satises

x∗n(x m) =δ mn

1.2 Banach space-valued random elements

We also dene the sequence of partial sum operators (for the Schauder basis(xn)) (Un) on E by

Un =

nXk=1

x∗k(x)xk =

nXk=1

A map T: E → F is called an isometry if

T(x)− T(y)

2=kx − yk1

for all x, y ∈ E Two normed spaces E and F are said to be isometric if there is

an isometry from each onto the other

Theorem 1.1.10 ([3]) Every separable normed space is isometric to a subspace

of C[0; 1]

1.2 Banach space-valued random elements

Let (Ω, A,P) be a probability space Let (E, k.k) be a real separable Banachspace We denote B(E) the Borel σ-algebra ofE, i.e., the σ-algebra generated bythe open sets in E

Denition 1.2.1 A map X: Ω → E is called a random element in E if it is(A, B(E))-measurable, i.e., X−1(B) ∈ A for all B ∈ B(E)

A random element X in E is called a countably-valued random element if it

is of the form

X =Xi∈I

xiIAi,

1.2 Banach space-valued random elements

where (xi) is a sequence inE, (Ai) is a disjoint sequence in A and I is countable

If I is nite, then X is called a simple random element

Below are some properties of random elements The reader may refer to [8]for the proofs of these properties

Proposition 1.2.2 ([8]) Let F be a real separable Banach space If X is arandom element in E and f: E → F is (B(E), B(F))-measurable map, then f(X)

Proposition 1.2.6 ([8]) X is a random element in E if and only if f(X) is arandom variable for each f ∈ E∗

Proposition 1.2.7 ([8]) If X is a random element in E and ξ is a randomvariable then ξX is a random element

Denition 1.2.8 The random elementsX1andX2inE are said to be identicallydistributed if

P[X1 ∈ B] = P[X2 ∈ B]for eachB ∈ B(E) A family of random elements is identically distributed if everypair is identically distributed

Denition 1.2.9 A nite set of random elements {X1, , X2} in E is said to

be independent if

P[X1 ∈ B1, , Xn ∈ Bn] = P[X1∈ B1] .P[Xn ∈ Bn]for every B1, , Bn ∈ B(E) A family of random elements in E is said to beindependent if every nite subset is independent

Proposition 1.2.10 ([8]) LetF be a real separable Banach space LetX1 andX2

be independent and identically distributed random elements inE and letf: E → F

be a measurable map Then f(X1) and f(X2) are independent and identicallydistributed random variables

1.3 Expectation

Proposition 1.2.11 ([8]) The random elements X1 and X2 in E are identicallydistributed if and only if f(X1) and f(X2) are identically distributed randomvariables for each f ∈ E∗

Proposition 1.2.12 ([8]) Let X1 and X2 be random elements in E Then X1

and X2 are independent if and only if f(X1) and g(X2) are independent randomvariables for every f, g ∈ E∗

Denition 1.2.13 A sequence of random elements (X n) is said to converge tothe random element X

(i) in probability if for each ε >0

limn→∞P [kXn− Xk > ε] = 0;

(ii) with probability 1 (w.p.1), or almost surely (a.s.) if

P

hlimn→∞ kXn− Xk > εi= 0.

If the sequence of random elements (Xn) converges to the random element X

in probability, we write

Xn − → X.P

If the sequence of random elements (Xn) converges to the random element X

with probability 1, we write

f(X) dP.

The element m is called the Pettis integral of X with respect to measure P Theelement m is also called the expectation or the mean of X and is denoted byZ

Ω

XdP or EX

Proposition 1.3.1 ([8]) Let X, X1, X2 be random elements in E and letx ∈ E.(i) If EX exists, then it is uniquely determined

1.4 Conditional probability, conditional expectations and martingales

(ii) If EX1 and EX2 exist, then E[X1+X2] = EX1+ EX2

(iii) If EX exists and a is a real number, then EaX =aEX

(iv) If P[X =x] = 1, then EX =x

(v) If Λ is a continuous linear map from E into a real separable Banach space

F and if EX exists, then EΛ(X) = Λ(EX)

(vi) If EX exists, then kEXk ≤EkXk.

A sucient condition for the existence of the expectation of a random element

is given in the following

Proposition 1.3.2 ([8]) LetX be a random element in the real separable Banachspace E If EkXk < ∞, then EX exists

Since, kXk is a real random variable if X is a random element in E, we havethe following Markov's inequality

P [kXk ≥ ε]≤ EkXk

r

ε rfor each ε >0and r >0, whenever the expectation EkXkr exists

Now, we introduce some terminologies which will be used later

Denition 1.3.3 Let X be a random element in E and p > 0

(i) EkXkp =

ZΩ

kXkpdP is called the p th moment of X

(ii) A random element in E is said to have the strong order p, or nite pthmoment, if EkXkp< ∞

(iii) A random element in E is said to have the weak order p if E f(X)

p

< ∞

for each f ∈ E∗

1.4 Conditional probability, conditional

expecta-tions and martingales

Let (Ω, A,P) be a probability space and let E be a real separable Banachspace Let A1 be a sub σ-algebra of A, we denote Lp(A1, E), 1≤ p < ∞, the set

1.4 Conditional probability, conditional expectations and martingales

of A1-measurable random elements in E having the strong order p and equip itwith the norm k.kp dened by

kXkp =

ZΩ

kXkpdP

1p

where X ∈ Lp(A1, E)

Let A1 be a sub σ-algebra of A and P1 is the restriction of P onto A1

Denition 1.4.1 The conditional probability with respect to σ-algebra A1 is themap

ZB

EA1X
dP1 =

ZB

XdP.

Based on the above proposition, the following denition is given

Denition 1.4.3 The operator EA 1: L1(A, E)→ L1(A1, E)is called the operator

of conditional expectation with respect to σ-algebra A1

The random element EA 1 X is called the conditional expectation of the randomelement X ∈ L1(A, X) with respect to σ-algebra A1

Below are some properties of conditional expectation

Proposition 1.4.4 ([1]) Let X and Y be random elements in E, a ∈ R.

(i) EA X =X; if A1 ={∅; Ω}, then EA 1 X = EX

(ii) If X ∈ L1(A1, E), then EA 1 X =X

(iii) If A1 ⊂ A2 are two sub σ-algebra of A, then EA 1

EA2X = EA1 X

1.4 Conditional probability, conditional expectations and martingales

(iv) If ξ is an A1-measurable random variable with E|ξ| < ∞ and EkξXk < ∞,then EA 1(ξX) =ξEA1 X; in particular, E ξEA1 X
= E(ξX)

(v) If A1 is generated by the partition {A1, , An}, then

EA1X =

nXk=1

1P(Ak)

(vii) If ϕ: E →R is a continuous convex functional and ϕ ◦ X ∈ L1(A,R), then

ϕ EA1 X
≤EA1 ϕ(X).

(viii) If X ∈ Lp(A, E), 1≤ p < ∞, then EA 1 X ∈ Lp(A1, E) EA1 X

p ≤ kXkp.Denition 1.4.5 Let (An)n≥1 be an increasing sequence of sub σ-algebras of

A and let (Xn)n≥1 be a sequence in L1(A, E) The sequence (Xn)n≥1 is called amartingale with respect to (An)n≥1 if for each n ≥1, Xn is An-measurable and

Xn = EAn Xm

for all m > n

CHAPTER 2

WEAK LAWS OF LARGE NUMBERS FOR

BANACH SPACE-VALUED RANDOM ELEMENTS

2.1 Denition and some well-known classical

re-sults

Let E be a real separable Banach space Let (Xn) be a sequence of randomelements inE We consider the elementSn which are certain specied symmetricfunctions of the rst n random elements of the sequence (X n)

S n =f n(X 1 , , X n).

If there exists a sequence (an) of elements of E such that for each ε >0

limn→∞P [kSn− ank > ε] = 0, (2.1)then the sequence (Xn) is said to obey the law of large numbers in the sense of(2.1)

Ordinarily, we often deal with the case when

S n = 1

n1p

nXk=1

"

> ε

#

= 0.

2.1 Denition and some well-known classical results

Proof Since the random elementsXn have nite variances, there exists a constant

1

n

nXk=1

Xk− 1n

nXk=1

EXk

1

n

nXk=1

Xk− 1n

nXk=1

EXk

> ε

#

≤0.

Since the probability cannot be less than zero, the theorem thus follows

Theorem 2.1.2 (Khinchin, [5]) If (Xn) is a sequence of independent and tically distributed random variables with EXn = a < ∞ for all n, then, for each

iden-ε >0,

limn→∞P

"

... class="text_page_counter">Trang 14

CHAPTER 2

WEAK LAWS OF LARGE NUMBERS FOR

BANACH SPACE-VALUED RANDOM ELEMENTS

2.1... the sequence (Xn) is said to obey the law of large numbers in the sense of( 2.1)

Ordinarily, we often deal with the case when

S n = 1... the existence of the expectation of a random element

is given in the following

Proposition 1.3.2 ([8]) LetX be a random element in the real separable Banachspace E