Almost sure central limit theorem

MINISTRY OF EDUCATION AND TRAINING HANOI PEDAGOGICAL UNIVERSITY DEPARTMENT OF MATHEMATICS PHAM NGOC QUYNH HUONG ALMOST SURE CENTRAL LIMIT THEOREM BACHELOR THESIS Hanoi, 2019 MINISTRY OF EDUCATION AND TRAINING HANOI PEDAGOGICAL UNIVERSITY DEPARTMENT OF MATHEMATICS PHAM NGOC QUYNH HUONG ALMOST SURE CENTRAL LIMIT THEOREM Speciality: Applied Mathmatics BACHELOR THESIS Supervisor: PhD Pham Viet Hung Hanoi, 2019 Confirmation This dissertation has been written on the basis of my research project carried at Hanoi Pedagogical University 2, under the supervision of PhD Pham Viet Hung The manuscript has never been published by others The author Pham Ngoc Quynh Huong Acknowledgment First and foremost, my heartfelt goes to my admirable supervisor, Mr Pham Viet Hung (Institute of Mathematics, Vietnam Academy of Science and Technology), for his continuous supports when I met obstacles during the journey The completion of this study would not have been possible without his expert advice, close attention and unswerving guidance Secondly, I am keen on expressing my deep gratitude for my family for encouraging me to continue this thesis I owe my special thanks to my parents for their emotional and material sacrifices as well as their understanding and unconditional support Finally, I own my thanks to many people who helped me and encouraged me during my work My special thanks to Mr Nguyen Phuong Dong (Hanoi Pedagogical University No.2) for his guidance on drawing up my work I am specially thankful to all my best friends at university for endless incentive Contents Preliminaries 1.1 1.2 Pr ob Ra nd 1.2 1.2 1.2 1.2 1.2 1.2 1.3 Ra 1.3 nd 1.3 1.3 1.4 Co 1.4 nv 1.4 1.4 1.4 1.4 1.5 Ce ntr 1.5 1.5 1.5 Almost sure central 3 4 14 41 41 16 61 61 71 92 22 22 42 26 2.1 Introduction 26 2.2 Almost Sure Central Limit Theorem 31 2.3 A universal result in Almost Sure Central Limit Theorem 36 References 37 limit theorem Introduction The Central Limit Theorem has been described as one of the most remarkable results, a beautiful pearl in a lot aspects of mathematics, especially in the world of probability and statistics It is one of the oldest results in probability theory, occupies a unique position at the heart of probabilistic limit theory and plays a central role in the theory of statistical inference Much of its importance stems from its proven adaptability and utility in many areas of mathematics, while it accounts hugely for the importance of the normal distribution in theoretical investigations The Central limit theorem has been developed in some directions to construct Kolmogorov distance, Cramer’s theorem (Large deviations) and so on In this thesis, a new direction to develop the theorem which was introduced in 1980s is presented: Almost Sure Central Limit Theorem The theorem is synthesized from the interest whether assertions are possible for almost every realization of the random variables Xn The Almost sure central limit theorem states in its simplest form that a sequence of independent, identically distributed random variables (Xn )n≥ , with E(X i) = and E(X 2i ) = 1, obeys ( ) N X P lim I √Sn ≤x = Φ(x) = 1, N →∞ log N n n=1 for each value x ∈ R I{.} here denotes the indicator function of events, Φ denotes the distribution function of the standard normal distribution and Sn is the nth partial sum of the above mentioned sequence of random variables This thesis mainly discuss about how the theorem was established, its statement and proof Besides, an universal result of this generalization is also under consideration Chapter Preliminaries 1.1 Probability Space Let Ω be a non-empty set without any special structure and 2Ω be the set of all subsets of Ω, including the empty set ∅ Definition 1.1.1 Let A be a subset of 2Ω Then A is a σ-algebra if it satisfies following properties: ∅ ∈ A and Ω ∈ A If A ∈ A then Ac := Ω \ A ∈ A A is closed under countable S∞ unions and countable intersections, i.e, if Ai ∈ A T∞ then i=1 Ai ∈ A and i=1 Ai ∈ A Definition 1.1.2 A probability measure defined on a σ-algebra A of Ω is a function P : A → [0, 1] which satisfies two following properties: P(Ω) = P possesses countable additivity, that is, for every countable sequence (An )n≥1 of elements of A, pairwise disjoint, one gets ! ∞ [∞ X P An = P(An ) n=1 n=1 Then, we have the definition of a probability space as follows Definition 1.1.3 The probability space (Ω, A, P) is constructed by three elements: the sample space Ω, σ-algebra A, probability measure P defined above Proposition 1.1.1 Let (Ω, A, P) be a probability space, then it has following proper- ties: (i) P(∅) = (ii) P(Ac ) = − P(A) (iii) P is additive (iv) If A, B ∈ A and A ⊆ B then P(A) ≤ P(B) BACHELOR THESIS PHAM NGOC QUYNH HUONG 1.2 Random Variables 1.2.1 Definition Definition 1.2.1 Let (Ω, A) be a measurable space and B(R) be the Borel σ-algebra on R A map X : Ω → R is said to be A-measurable if X −1 (B) := {ω : X(ω) ∈ B} ∈ A, ∀B ∈ B(R) Then, the A-measurable function X is called a random variable Note that there are types of random variables: discrete random variables and continuous random variables A discrete random variable has a finite or countable range whereas a continuous function takes on an uncountably infinite number of possible outcomes Remark 1.2.1 X is a random variable if and only if for all a ∈ R, {ω : X(ω) < a} ∈ A Remark 1.2.2 Let ϕ : R → R be a measurable function Then ϕ(X) is also a random variable 1.2.2 Distribution functions Definition 1.2.2 The function FX : R → R that satisfies FX (x) = P[X < x], x∈R is called the distribution function of X Besides, one can check that FX has the following properties: FX is a non-decreasing function; F is left-continuous and has right limit at any point in R; lim FX (x) = and lim FX (x) = x→−∞ x→∞ Definition 1.2.3 Let X be a continuous random variable If there exists a function fX satisfying Za FX (a) = P[X < a] = fX (x)dx ∀ a ∈ R −∞ then one says fX to be the density function of X In addition, the density function f = fX has properties as follows: Z +∞ f (x) is non-negative for all x ∈ R and f (x)dx = −∞ Z b f (x)dx for all x ∈ R and for any a, b ∈ R such that a < b P[a < X < b] = a Moreover, for any A ∈ B(R), it is true that Z P[X ∈ A] = f (x)dx A Here I present some celebrated kinds of distributions for discrete random variables and continuous random variables: Example 1.2.1 (For discrete random variables) Poisson distribution X is said to have Poisson distribution with parameter λ > 0, denoted by X ∼ P oi(λ) if X(Ω) = {0, 1, 2, } and e−λ λk k!, k = 0, 1, Bernoulli distribution X has Bernoulli distribution with parameter p ∈ [0, 1] if it takes only and as its range and P[X = k] = P[X = 1] = − P[X = 0] = p X represents a kind of experiments with only two outcomes: ”success” (X = 1) and failure (X = 0) Binomial distribution X has Binomial distribution with parameter p ∈ [0, 1] and n ∈ N, denoted by X ∼ B(n, p), if X takes values in {0, 1, , n} and n k p (1 − p)n−k , P[X = k] = k where k = 0, 1, , n Example 1.2.2 (For continuous random variables) Uniform distribution The function   if a ≤ x ≤ b f (x) = b−a 0 otherwise is called the Uniform distribution on [a, b] and denoted by U [a, b].The distribution function corresponding to f is  if x < a 0  x−a F (x) = b−a if a ≤ x ≤ b  if x > b Normal distribution The Normal distribution with the mean a and variance σ has the form (x−a)2 f −(x) = √ 2σ , x ∈ R, e 2πσ If a = and σ = 1, N (0, 1) is the standard normal distribution 62 63 64 65 66 67 68 69 70 71 72 73 74 75 ϕ0 76 ... 2.2 Almost Sure Central Limit Theorem 31 2.3 A universal result in Almost Sure Central Limit Theorem 36 References 37 limit theorem. .. presented: Almost Sure Central Limit Theorem The theorem is synthesized from the interest whether assertions are possible for almost every realization of the random variables Xn The Almost sure central. .. The Central limit theorem has been developed in some directions to construct Kolmogorov distance, Cramer’s theorem (Large deviations) and so on In this thesis, a new direction to develop the theorem