Luận văn toán học DOMAINS OF PARTIAL ATTRACTION OF INFINITELY DIVISIBLE PROBABILITY MEASURES

VIETNAM NATIONAL UNIVERSITYUNIVERSITY OF SCIENCEFACULTY OF MATHEMATICS, MECHANICS AND INFORMATICSTran Anh Chinh DOMAINS OF PARTIAL ATTRACTION OF INFINITELY DIVISIBLE PROBABILITY MEASURES

VIETNAM NATIONAL UNIVERSITYUNIVERSITY OF SCIENCEFACULTY OF MATHEMATICS, MECHANICS AND INFORMATICS

Tran Anh Chinh

DOMAINS OF PARTIAL ATTRACTION OF INFINITELY

DIVISIBLE PROBABILITY MEASURES

Undergraduate Thesis Advanced Undergraduate Program in Mathematics

Hanoi - 2012

VIETNAM NATIONAL UNIVERSITYUNIVERSITY OF SCIENCEFACULTY OF MATHEMATICS, MECHANICS AND INFORMATICS

Tran Anh Chinh

DOMAINS OF PARTIAL ATTRACTION OF INFINITELY

DIVISIBLE PROBABILITY MEASURES

Undergraduate Thesis Advanced Undergraduate Program in Mathematics

Thesis advisor: Assoc.Prof.Dr Ho Dang Phuc

Hanoi - 2012

I want to express my sincere gratitude to my thesis advisor Ass Prof Dr HoDang Phuc, who has introduced me to the field of Probability I am especially gratefulfor your continual availability for discussions, your patience and ability of makingabstract mathematics so easy to be perceived You are such a wonderful and greatthesis advisor

Great deals appreciated go to the contribution of my faculty of Mathematics Mechanics - Informatics in my five years education process That provide me knowledge

-in mathematics

Last but not least, the thanks are directed to my family, especially to my parentsfor their instant support and encouragement across all the years of my study in theuniversity

Tran Anh Chinh

Ha Noi, December, 2012

The second chapter, we review some important knownlege on infinitely divisibledistribution functions and its two special cases of stable distributions and semi-stabledistributions In additions, the canonical representations of those three kinds of prob-ability distributions are given.

In the third chapter, we study properties of domains of partial attraction, domains

of semi-attraction, and domains of attraction for probability measures Some relationsbetween their domains and stability, semi-stability and infinite divisibility of probabil-ity measures are discussed

tions 162.3.2 Canonical representation of stable characteristic functions 232.3.3 Canonical representation of semi-stable characteristic functions 30

3 Domains of partial attraction, domains of semi-attraction, and

3.1 Domains of partial attraction and universal distribution 343.2 Domains of attraction and domains of semi-attraction 44

Chapter 1

Basic concepts

1.1 Convergence of distribution function

Let DF be classes of distribution functions (d.f) on R, that means F ∈ DF ifa) 0 ≤ F (x) ≤ 1, ∀x ∈ R;

Example 1.1.2 Let F2(x) satisfy:

F2(x) =n

0, x ≤ −11/2, −1 < x ≤ 1

1, x > 1Example 1.1.3 Let F3(x) satisfy

Let (Ω,F, P) be a probability space Let ξ be a random variable (r.v) on R, thatmeans ξ : Ω → R, and ξ be measurable Then it is clear that the function

Fξ(x) = P{ω : ξ(ω) < x}

be a distribution function Fξ is called the d.f of random variable ξ

We can see that F1 is distribution function of the improper random variable takenvalue equal x0 with probability 1, F2 is the d.f of the Bernoulli taken value -1 and +1,each with probability 1/2; F3 is the distribution function of standard normal randomvariable

Definition 1.1.1 (Weak convergence)

Let {Fn} be a sequence of d.f’s; Fn, F ∈ DF {Fn} is said weakly converges to bution function F if

distri-Fn(x) → F (x), n → ∞, ∀x ∈ C(F ),where C(F ) is set of continuity points of F , and we denote: Fn⇒ F

In the next theorem, we can see that the weakly convergence of distribution tions is equivalent to the convergence of integral sequences R∞

func-−∞g(x)dFn(x), for allbounded and continuous function g on R

Proof See in [2] page 197 to 202

Moreover, let (Pn) and P are probability measures on (R, BR) For BR is the Borelσ-field of R and elements of BR are called Borel sets And of course BR satisfies thefollowing condition:

(a) Pn⇒ P;

(b) limPn(A) ≤ P(A) for every closed set A;

(c) limPn(A) ≥ P(A) for every open set A;

(d) lim Pn(A) = P(A) for every Borel set A whose boundary has P-measure 0.Proof See [11] page 172-174

Definition 1.1.2 LetP denote the family of all probability measures on R A family ofprobability measures M ⊂P is called relative compact if for every sequence (Pn) ⊂ Mthere exists a subsequence (Pn k) weakly convergent

Definition 1.1.3 A family of probability measure {Pα, α ∈ M } ⊂P is called tight iffor every  > 0 there exists a compact set K ⊂ R such that

sup

α Pα(R − K) < 

A family of distribution function {Fα, α ∈ M } on Rd is tight if family of correspondingprobability function is tight

Proposition 1.1.2 ( Prokhorov theorem )

A family of probability measures on R is relative compact if and only if it is tight.Proof See [11] page 179-182

Proposition 1.1.3 Suppose (i) that A is the subclass of BR, A is closed under theformation of infinite intersections (π-system) and (ii) for every x in BR and positive

, there is in A an A for which x ∈ A0

⊂ A ⊂ B(x, ) If Pn(A) → P(A) for every A

in A, then Pn⇒ P

Proof See [6] page 17

Proposition 1.1.4 Let S ⊂ R be a dense subset of R in S Then the set of allmeasures whose supports are finite subsets of S is dense in P

Proof See [5] page 44-45

is called characteristic function (c.f) of F

If F is distribution function of random variable ξ, then the corresponding teristic function is called the characteristic function of random variable ξ In that case,

charac-we see that

f (t) = Eeitξ.Proposition 1.2.1

a) f (0) = 1 and |f (1)| ≤ 1, ∀t ∈ R;

b) f (−t) = f (t);

c) f (t) is uniformly continuous on R

d) For all real numbers a and b, we have

faξ+b(t) = eibtfξ(at)

e) If ξ and η is independent random variable, then

fξ+η(t) = fξ(t).fη(t)

Proof a)+b)+c)

Part a) and b) follows immediately from definition of a characteristic function(f (t) = Eeitξ) It remains to prove the uniform continuity of the function For thispurpose we introduce an inequality which will be useful also in what follows,namely,if

F (A) − F (−A) ≥ 1 − ,(Let A be finite number) then

|eiz00x− eiz0x|dF (x)

≤Z

Example 1.2.2 Let X has Poisson distribution with λ > 0, fX is characteristicfunction of X,

−λ

.eλeit = eλ(eit−1)

In the next theorems, we can see two main properties of relationship between acteristic function (c.f) and distribution function, that are: unique determined, andcorresponding of continuity between c.f and d.f

char-Theorem 1.2.1 Let F be a distribution function, f is its characteristic function If

x1, x2 is two continuity points of F (x), then

Z Z c 0

hsin t(z − x1)

t − sin t(z − x2)

t

idtdF (z)

Now, for every α and c

1π

Z c o

sin αt

t dt

=

1π

Z αc 0

sin s

s ds

sin τ x

τ x

≤ 1,

sin τ x

τ x

≤ 1

τ x.

It is easy to see that (1.7) is equivalents to (1.5)

Now let the conditions of this theorem satisfied Then for every  > 0 we can find

a τ > 0 such that

12τ

Z +τ

−τ

f (t)dt − 1

... 1, f or x → +∞,

be satisfied uniformly in S

Proof (of Theorem 1.2.3 )

Let Fn ⇒ F From definition of weak convergence it follows that fn(t) → f (t)... uniquely determined by its characteristicfunction

Proof From above theorem it follows immediately that at every continuity points x ofthe function F (x) the following formula applies:

F... proposition:

Proposition 1.2.2 In the order that the set S of distributions be conditionally pact ( which mean for every sequences of family S there exists subsequences is weaklyconvergence )