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Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2011, Article ID 576301, 9 pages doi:10.1155/2011/576301 Research Article Almost Sure Central Limit Theorem for Product of Partial Sums of Strongly Mixing Random Variables Daxiang Ye and Qunying Wu College of Science, Guilin University of Technology, Guilin 541004, China Correspondence should be addressed to Daxiang Ye, 3040801111@163.com Received 19 September 2010; Revised 1 January 2011; Accepted 26 January 2011 Academic Editor: Ond ˇ rej Do ˇ sl ´ y Copyright q 2011 D. Ye and Q. Wu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We give here an almost sure central limit theorem for product of sums of strongly mixing positive random variables. 1. Introduction and Results In recent decades, there has been a lot of work on the almost sure central limit theorem ASCLT, we can refer to Brosamler 1, Schatte 2, Lacey and Philipp 3, and Peligrad and Shao 4. Khurelbaatar and Rempala 5 gave an ASCLT for product of partial sums of i.i.d. random variables as follows. Theorem 1.1. Let {X n ,n ≥ 1} be a sequence of i.i.d. positive random variables with EX 1 μ>0 and VarX 1 σ 2 . Denote γ σ/μ the coefficient of variation. Then for any real x lim n →∞ 1 ln n n k1 1 k I ⎛ ⎝ k i1 S i k!μ k 1/γ √ k ≤ x ⎞ ⎠ F x a.s., 1.1 where S n n k1 X k , I∗ is the indicator function, F· is the distribution function of the random variable e N , and N is a standard normal variable. Recently, Jin 6 had p roved that 1.1 holds under appropriate conditions for strongly mixing positive random variables and gave an ASCLT for product of partial sums of strongly mixing as follows. 2 Journal of Inequalities and Applications Theorem 1.2. Let {X n ,n ≥ 1} be a sequence of identically distributed positive strongly mixing random variable with EX 1 μ>0 and VarX 1 σ 2 , d k 1/k, D n n k1 d k . Denote by γ σ/μ the coefficient of variation, σ 2 n Var n k1 S k − kμ/kσ and B 2 n VarS n . Assume E | X 1 | 2δ < ∞ for some δ>0, lim n →∞ B 2 n n σ 2 0 > 0, α n O n −r for some r>1 2 δ , inf n∈N σ 2 n n > 0. 1.2 Then for any real x lim n →∞ 1 D n n k1 d k I ⎛ ⎝ k i1 S i k!μ k 1/γσ k ≤ x ⎞ ⎠ F x a.s. 1.3 The sequence {d k ,k ≥ 1} in 1.3 is called weight. Under the conditions of Theorem 1.2, it is easy to see that 1.3 holds for every sequence d ∗ k with 0 ≤ d ∗ k ≤ d k and D ∗ n k≤n d ∗ k →∞ 7. Clearly, the larger the weight sequence d k is, the stronger is the result 1.3. In the following sections, let d k e ln k α /k,0≤ α<1/2,D n n k1 d k ,“” denote the inequality “≤” up to some universal constant. We first give an ASCLT for strongly mixing positive random variables. Theorem 1.3. Let {X n ,n ≥ 1} be a sequence of identically distributed positive strongly mixing random variable with EX 1 μ>0 and VarX 1 σ 2 , d k and D n as mentioned above. Denote by γ σ/μ the coefficient of variation, σ 2 n Var n k1 S k − kμ/kσ and B 2 n VarS n . Assume that E | X 1 | 2δ < ∞ for some δ>0, 1.4 α n O n −r for some r>1 2 δ , 1.5 lim n Central Limit Theorem (Cookie Recipes) Central Limit Theorem (Cookie Recipes) By: OpenStaxCollege Central Limit Theorem (Cookie Recipes) Class Time: Names: Student Learning Outcomes • The student will demonstrate and compare properties of the central limit theorem GivenX = length of time (in days) that a cookie recipe lasted at the Olmstead Homestead (Assume that each of the different recipes makes the same quantity of cookies.) Recipe # X Recipe # X Recipe # X Recipe # X 1 16 31 46 2 17 32 47 18 33 48 11 19 34 49 5 20 35 50 21 36 51 22 37 52 23 38 53 24 39 54 10 25 40 55 11 26 41 56 1/4 Central Limit Theorem (Cookie Recipes) Recipe # X Recipe # X Recipe # X Recipe # X 12 27 42 57 13 28 43 58 14 29 44 59 15 30 45 60 Calculate the following: μx = _ σx = _ Collect the DataUse a random number generator to randomly select four samples of size n = from the given population Record your samples in [link] Then, for each sample, calculate the mean to the nearest tenth Record them in the spaces provided Record the sample means for the rest of the class Complete the table: Sample Sample Sample Sample Sample means from other groups: ¯ Means: x = ¯ x= ¯ x= ¯ x= Calculate the following: ¯ x = _ s¯x = _ Again, use a random number generator to randomly select four samples from the population This time, make the samples of size n = 10 Record the samples in [link] As before, for each sample, calculate the mean to the nearest tenth Record them in the spaces provided Record the sample means for the rest of the class Sample Sample Sample Sample Sample means from other groups 2/4 Central Limit Theorem (Cookie Recipes) Sample Sample Sample Sample Sample means from other groups ¯ Means: x = ¯ x= ¯ x= ¯ x= Calculate the following: ¯ x = s¯x = For the original population, construct a histogram Make intervals with a bar width of one day Sketch the graph using a ruler and pencil Scale the axes Draw a smooth curve through the tops of the bars of the histogram Use one to two complete sentences to describe the general shape of the curve Repeat the Procedure for n = For the sample of n = days averaged together, construct a histogram of the averages (your means together with the means of the other groups) Make intervals with bar widths of a day Sketch the graph using a ruler and pencil Scale the axes 3/4 Central Limit Theorem (Cookie Recipes) Draw a smooth curve through the tops of the bars of the histogram Use one to two complete sentences to describe the general shape of the curve Repeat the Procedure for n = 10 For the sample of n = 10 days averaged together, construct a histogram of the averages (your means together with the means of the other groups) Make intervals with bar widths of a day Sketch the graph using a ruler and pencil Scale the axes Draw a smooth curve through the tops of the bars of the histogram Use one to two complete sentences to describe the general shape of the curve Discussion Questions Compare the three histograms you have made, the one for the population and the two for the sample means In three to five sentences, describe the similarities and differences State the theoretical (according to the clt) distributions for the sample means ¯ n = 5: x ~ _( _, _) ¯ n = 10: x ~ _( _, _) Are the sample means for n = and n = 10 “close” to the theoretical mean, μx? Explain why or why not Which of the two distributions of sample means has the smaller standard deviation? Why? As n changed, why did the shape of the distribution of the data change? Use one to two complete sentences to explain what happened 4/4 Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010, Article ID 130915, 10 pages doi:10.1155/2010/130915 Research Article Almost Sure Central Limit Theorem for a Nonstationary Gaussian Sequence Qing-pei Zang School of Mathematical Science, Huaiyin Normal University, Huaian 223300, China Correspondence should be addressed to Qing-pei Zang, zqphunhu@yahoo.com.cn Received 4 May 2010; Revised 7 July 2010; Accepted 12 August 2010 Academic Editor: Soo Hak Sung Copyright q 2010 Qing-pei Zang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Let {X n ; n ≥ 1} be a standardized non-stationary Gaussian sequence, and let denote S n n k1 X k , σ n VarS n . Under some additional condition, let the constants {u ni ;1≤ i ≤ n, n ≥ 1} satisfy n i1 1 −Φu ni → τ as n →∞for some τ ≥ 0andmin 1≤i≤n u ni ≥ clog n 1/2 ,forsomec>0, then, we have lim n →∞ 1/ log n n k1 1/kI{∩ k i1 X i ≤ u ki ,S k /σ k ≤ x} e −τ Φx almost surely for any x ∈ R,whereIA is the indicator function of the event A and Φx stands for the standard normal distribution function. 1. Introduction When {X, X n ; n ≥ 1} is a sequence of independent and identically distributed i.i.d. random variables and S n n k1 X k ,n ≥ 1,M n max 1≤k≤n X k for n ≥ 1. If EX0, VarX1, the so-called almost sure central limit theorem ASCLT has the simplest form as follows: lim n →∞ 1 log n n k1 1 k I S k √ k ≤ x Φ x , 1.1 almost surely for all x ∈ R, where IA is the indicator function of the event A and Φx stands for the standard normal distribution function. This result was first proved independently by Brosamler 1 and Schatte 2 under a stronger moment condition; since then, this type of almost sure version was extended to different directions. For example, Fahrner and Stadtm ¨ uller 3 and Cheng et al. 4 extended this almost sure convergence for partial sums to the case of maxima of i.i.d. random variables. Under some natural conditions, they proved as follows: lim n →∞ 1 log n n k1 1 k I M k − b k a k ≤ x G x a.s. 1.2 2 Journal of Inequalities and Applications for all x ∈ R, where a k > 0andb k ∈ R satisfy P M k − b k a k ≤ x −→ G x , as k −→ ∞ 1.3 for any continuity point x of G. In a related work, Cs ´ aki and Gonchigdanzan 5 investigated the validity of 1.2 for maxima of stationary Gaussian sequences under some mild condition whereas Chen and Lin 6 extended it to non-stationary Gaussian sequences. Recently, Dudzi ´ nski 7 obtained two-dimensional version for a standardized stationary Gaussian sequence. In this paper, inspired by the above results, we further study ASCLT in the joint version for a non-stationary Gaussian sequence. 2. Main Result Throughout this paper, let {X n ; n ≥ 1} be a non-stationary standardized normal sequence, and σ n VarS n .Herea b and a ∼ b stand for a Ob and a/b → 1, respectively. Φx is the standard normal distribution function, and φx is its density function; C will denote a positive constant although its value may change from one appearance to the next. Now, we state our main result as follows. Theorem 2.1. Let {X n ; n ≥ 1} be a sequence of non-stationary standardized Gaussian variables with covariance matrix r ij such that 0 ≤ r ij ≤ ρ |i−j| for i / j,whereρ n ≤ 1 for all n ≥ 1 and sup s≥n s−1 is−n ρ i log Vietnam Journal of Mathematics 33:4 (2005) 443–461 Central Limit Theorem for Functional of Jump Markov Processes Nguyen Van Huu, Vuong Quan Hoang, and Tran Minh Ngoc Department of Mathematics Hanoi National University, 334 Nguyen Trai Str., Hanoi, Vietnam Received February 8, 2005 Revised May 19, 2005 Abstract. In this paper some conditions a re given to ensure that for a jump homoge- neous Markov process {X(t),t≥ 0} the law of the integral functional of the process: T −1/2 T 0 ϕ(X(t))dt, converges to the normal law N(0,σ 2 ) as T →∞,whereϕ is a mapping from the state space E into R. 1. Introduction The central limit theorem is a subject investigated intensively by many well- known probabilists such as Linderberg, Chung, The results concerning cen- tral limit theorems, the iterated logarithm law, the lower and upper bounds of the moderate deviations are well understood for independent random variable sequences and for martingales but less is known for dependent random variables such as Markov chains and Markov processes. The first result on central limit for functionals of stationary Markov chain with a finite state space can be found in the book of Chung [5]. A technical method for establishing the central limit is the regeneration method. The main idea of this method is to analyse the Markov process with arbitrary state space by dividing it into independent and identically distributed random blocks between visits to fixed state (or atom). This technique has been developed by Athreya - Ney [2], Nummelin [10], Meyn - Tweedie [9] and recently by Chen [4]. The technical method used in this paper is based on central limit for mar- tingales and ergodic theorem. The paper is ogranized as follows: In Sec. 2, we shall prove that for a positive recurrent Markov sequence 444 Nguyen Van Huu, Vuong Quan Hoang, and Tran Minh Ngoc {X n ,n≥ 0} with Borel state space (E, B)andforϕ : E → R such that ϕ(x)=f(x) −Pf(x)=f(x) − E f(y)P (x, dy) with f : E → R such that E f 2 (x)Π(dx) < ∞,whereP (x, .) is the transition probability and Π(.) is the stationary distribution of the process, the distribution of n −1/2 n i=1 ϕ(X i ) converges to the normal law N(0,σ 2 )withσ 2 = E (ϕ 2 (x)+ 2ϕ(x)Pf(x))Π(dx). The central limit theorem for the integral functional T −1/2 T 0 ϕ(X(t))dt of jump Markov process {X(t),t≥ 0} will be established and proved in Sec. 3. Some examples will be given in Sec. 4. It is necessary to emphasize that the conditions for normal asymptoticity of n −1/2 n i=1 ϕ(X i ) is the same as in [8] but they are not equivalent to the ones established in [10, 11]. The results on the central limit for jump Markov processes obtained in this paper are quite new. 2. Central Limit for the Functional of Markov Sequence Let us consider a Markov sequence {X n ,n ≥ 0} defined on a basic probability space (Ω, F,P) with the Borel state space (E,B), where B is the σ-algebra generated by the countable family of subsets of E. Suppose that {X n ,n≥ 0} is homogeneous with transition probability P (x, A)=P (X n+1 ∈ A|X n = x),A∈B. We have the following definitions Definition 2.1. Markov process {X n ,n ≥ 0} is said to be irreducible if there exists a σ- finite measure μ on (E, B) such that for all A ∈B μ(A) > 0 implies ∞ n=1 P n (x, A) > 0, ∀x ∈ E where P n (x, A)=P (X m+n ∈ A|X m = x). The measure μ is called irreducible measure. By Proposition 2.4 of Nummelin [10], there exists a maximum irreducible measure μ ∗ possessing the property that if μ is any irreducible measure then μ μ ∗ . Definition 2.2. Markov process {X n ,n≥ 0} is said to be recurrent if ∞ n=1 P n (x, A)=∞, ∀x ∈ E,∀A ∈B: μ ∗ (A) > 0. The process is said to be Harris recurrent if P x (X n ∈ Ai.o.)=1. Central Limit Theor em for Functional of Jump Markov Processes 445 Let us notice that a process which is Harris recurrent is also recurrent. Theorem 2.1. If {X n ,n≥ 0} is recurrent then there exists a uniquely invariant measure Π(.) on (E,B) (up to constant multiples) in CENTRAL LIMIT THEOREM OF LINEAR SPECTRAL STATISTICS FOR LARGE DIMENSIONAL RANDOM MATRICES WANG XIAOYING NATIONAL UNIVERSITY OF SINGAPORE 2009 CENTRAL LIMIT THEOREM OF LINEAR SPECTRAL STATISTICS FOR LARGE DIMENSIONAL RANDOM MATRICES WANG XIAOYING (B.Sc. Northeast Normal University, China) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF STATISTICS AND APPLIED PROBABILITY NATIONAL UNIVERSITY OF SINGAPORE 2009 ii Acknowledgements I would like to express my deep and sincere gratitude to my supervisors, Professor Bai Zhidong and Associate Professor Zhou Wang. Their valuable guidance and continuous support have been crucial to the completion of this thesis. I appreciate all the time and efforts they have spent in helping me to solve the problems I encountered. I have learned many things from them, especially regarding academic research and character building. Special acknowledgement are also due to Assistant Professor Pan Guangming and Mr. Wang Xiping for discussions on various topics of large dimensional random matrices theory. It is a great pleasure to record my thanks to my dear friends Ms. Zhao Wanting, Ms. Zhao Jingyuan, Ms. Zhang Rongli, Ms. Li Hua, Ms. Zhang Xiaoe, Ms. Li Xiang, Mr. Khang Tsung Fei, Mr. Li Mengxin, Mr. Deng Niantao, Mr. Su Yue, Mr. Wang Daqing, and Mr. Loke Chok Kang, who have given me much help not only in my study but also in my daily life. Sincere thanks to all my friends who helped me in one way or another iii for their friendship and encouragement. On a personal note, I thank my parents, husband, sisters and brother for their endless love and continuous support during the entire period of my PhD programme. I also thank my baby for giving me a lot of happy times and a sense of responsibility. Finally, I would like to attribute the completion of this thesis to other members and staff of the department for their help in various ways and providing such a pleasant studying environment. I also wish to express my gratitude to the university and the department for supporting me through an NUS research scholarship. iv Contents Contents Acknowledgements Summary ii vii Introduction 1.1 Large Dimensional Random Matrices . . . . . . . . . . . . . . . . . . 1.2 Spectral Analysis of LDRM . . . . . . . . . . . . . . . . . . . . . . . 1.3 Methodologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Moment Method . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Stieltjes Transform . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Orthogonal Polynomial Decomposition . . . . . . . . . . . . . 11 Organization of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . 12 1.4 v Contents Literature Review 14 2.1 Limiting Spectral Distribution (LSD) of LDRM . . . . . . . . . . . . . 14 2.1.1 Wigner Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.1.2 Sample Covariance Matrix . . . . . . . . . . . . . . . . . . . . 16 2.1.3 Product of Two Random Matrices . . . . . . . . . . . . . . . . 18 2.2 Limits of Extreme Eigenvalues . . . . . . . . . . . . . . . . . . . . . . 19 2.3 Convergence Rate of ESD . . . . . . . . . . . . . . . . . . . . . . . . 21 2.4 CLT of Linear Spectral Statistics (LSS) . . . . . . . . . . . . . . . . . 22 CLT of LSS for Wigner Matrices 26 3.1 Introduction and Main Result . . . . . . . . . . . . . . . . . . . . . . . 26 3.2 Bernstein Polynomial Approximation . . . . . . . . . . . . . . . . . . 29 3.3 Truncation and Preliminary Formulae . . . . . . . . . . . . . . . . . . 33 3.3.1 Simplification by Truncation . . . . . . . . . . . . . . . . . . 33 3.3.2 Preliminary Formulae . . . . . . . . . . . . . . . . . . . . . . 35 3.4 The Mean Function of LSS . . . . . . . . . . . . . . . . . . . . . . . . 36 3.5 Convergence of ∆ − E∆ . . . . . . . . . . . . . . . . . . . . . . . . . 49 Contents vi CLT of LSS for Sample Covariance Matrices 61 4.1 CONVERGENCE RATE IN THE CENTRAL LIMIT THEOREM FOR THE CURIE-WEISS-POTTS MODEL HAN HAN (HT080869E) A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF SCIENCE DEPARTMENT OF MATHEMATICS NATIONAL UNIVERSITY OF SINGAPORE i Acknowledgements First and foremost, it is my great honor to work under Assistant Professor Sun Rongfeng, for he has been more than just a supervisor to me but as well as a supportive friend; never in my life I have met another person who is so knowledgeable but yet is extremely humble at the same time. Apart from the inspiring ideas and endless support that Prof. Sun has given me, I would like to express my sincere thanks and heartfelt appreciation for his patient and selfless sharing of his knowledge on probability theory and statistical mechanics, which has tremendously enlightened me. Also, I would like to thank him for entertaining all my impromptu visits to his office for consultation. Many thanks to all the professors in the Mathematics department who have taught me before. Also, special thanks to Professor Yu Shih-Hsien and Xu Xingwang for patiently answering my questions when I attended their classes. I would also like to take this opportunity to thank the administrative staff of the Department of Mathematics for all their kindness in offering administrative assistant once to me throughout my master’s study in NUS. Special mention goes to Ms. Shanthi D/O D Devadas, Mdm. Tay Lee Lang and Mdm. Lum Yi Lei for always entertaining my request with a smile on their face. Last but not least, to my family and my classmates, Wang Xiaoyan, Huang Xiaofeng and Hou Likun, thanks for all the laughter and support you have given me throughout my master’s study. It will be a memorable chapter of my life. Han Han Summer 2010 Contents Acknowledgements i Summary iii 1 Introduction 1 2 The Curie-Weiss-Potts Model 4 2.1 The Curie-Weiss-Potts Model . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 The Phase Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3 Stein’s Method and Its Application 17 3.1 The Stein Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.2 The Stein Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.3 An Approximation Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.4 An Application of Stein’s Method . . . . . . . . . . . . . . . . . . . . . . . 23 4 Main Results 31 Bibliography 37 ii iii Summary There is a long tradition in considering mean-field models in statistical mechanics. The Curie-Weiss-Potts model is famous, since it exhibits a number of properties of real substances, such as multiple phases, metastable states and others, explicitly. The aim of this paper is to prove Berry-Esseen bounds for the sums of the random variables occurring in a statistical mechanical model called the Curie-Weiss-Potts model or mean-field Potts model. To this end, we will apply Stein’s method using exchangeable pairs. The aim of this thesis is to calculate the convergence rate in the central limit theorem for the Curie-Weiss-Potts model. In chapter 1, we will give an introduction to this problem. In chapter 2, we will introduce the Curie-Weiss-Potts model, including the Ising model and the Curie-Weiss model. Then we will give some results about the phase transition of the Curie-Weiss-Potts model. In chapter 3, we state Stein’s method first, then give the Stein operator and an approximation theorem. In section 4 of this chapter, we will give an application of Stein’s method. In chapter, we will state the main result of this thesis and prove it. Chapter 1 Introduction There is a long tradition in considering mean-field models in statistical mechanics. The Curie-Weiss-Potts model is famous, since it exhibits a number of properties of real substances, such as multiple phases, metastable states and others, explicitly. The aim of this paper is to prove Berry-Esseen bounds for the sums of the .. .Central Limit Theorem (Cookie Recipes) Recipe # X Recipe # X Recipe # X Recipe # X 12 27 42 57 13 28 43 58 14 29... rest of the class Sample Sample Sample Sample Sample means from other groups 2/4 Central Limit Theorem (Cookie Recipes) Sample Sample Sample Sample Sample means from other groups ¯ Means: x =... bar widths of a day Sketch the graph using a ruler and pencil Scale the axes 3/4 Central Limit Theorem (Cookie Recipes) Draw a smooth curve through the tops of the bars of the histogram Use one