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This Provisional PDF corresponds to the article as it appeared upon acceptance. Fully formatted PDF and full text (HTML) versions will be made available soon. A note on the almost sure limit theorem for self-normalized partial sums of random variables in the domain of attraction of the normal law Journal of Inequalities and Applications 2012, 2012:17 doi:10.1186/1029-242X-2012-17 Qunying Wu (wqy666@glite.edu.cn) ISSN 1029-242X Article type Research Submission date 4 August 2011 Acceptance date 20 January 2012 Publication date 20 January 2012 Article URL http://www.journalofinequalitiesandapplications.com/content/2012/1/17 This peer-reviewed article was published immediately upon acceptance. It can be downloaded, printed and distributed freely for any purposes (see copyright notice below). For information about publishing your research in Journal of Inequalities and Applications go to http://www.journalofinequalitiesandapplications.com/authors/instructions/ For information about other SpringerOpen publications go to http://www.springeropen.com Journal of Inequalities and Applications © 2012 Wu ; licensee Springer. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. A note on the almost sure limit theorem for self-normalized partial sums of random variables in the domain of attraction of the normal law Qunying Wu 1,2 1 College of Science, Guilin University of Technology, Guilin 541004, P. R. China 2 Guangxi Key Laboratory of Spatial Information and Geomatics, Guilin 541004, P.R. China Email address: wqy666@glite.edu.cn Abstract Let X, X 1 , X 2 , . . . be a sequence of independent and identically distributed random variables in the domain of attraction of a normal distribution. A universal result in almost sure limit theorem for the self-normalized partial sums S n /V n is established, where S n = n i=1 X i , V 2 n = n i=1 X 2 i . Mathematical Scientific Classification: 60F15. Keywords: domain of attraction of the normal law; self-normalized partial sums; almost sure central limit theorem. 1. Introduction Throughout this article, we assume {X, X n } n∈N is a sequence of independent and identically distributed (i.i.d.) random variables with a non-degenerate distribution function F. For each n ≥ 1, the symbol S n /V n denotes self- normalized partial sums, where S n = n i=1 X i , V 2 n = n i=1 X 2 i . We say that the random variable X belongs to the domain of attraction of the normal law, if there exist constants a n > 0, b n ∈ R such that S n − b n a n d −→ N, (1) where N is the standard normal random variable. We say that {X n } n∈N satisfies the central limit theorem (CLT). It is known that (1) holds if and only if lim x→∞ x 2 P(|X| > x) EX 2 I(|X| ≤ x) = 0. (2) In contrast to the well-known classical central limit theorem, Gine et al. [1] obtained the following self-normalized version of the central limit theorem: The Central Limit Theorem for Sums The Central Limit Theorem for Sums By: OpenStaxCollege Suppose X is a random variable with a distribution that may be known or unknown (it can be any distribution) and suppose: μX = the mean of Χ σΧ = the standard deviation of X If you draw random samples of size n, then as n increases, the random variable ΣX consisting of sums tends to be normally distributed and ΣΧ ~ N((n)(μΧ), (√n)(σΧ)) The central limit theorem for sums says that if you keep drawing larger and larger samples and taking their sums, the sums form their own normal distribution (the sampling distribution), which approaches a normal distribution as the sample size increases The normal distribution has a mean equal to the original mean multiplied by the sample size and a standard deviation equal to the original standard deviation multiplied by the square root of the sample size The random variable ΣX has the following z-score associated with it: Σx is one sum z = Σx – (n)(μX) (√n)(σX) (n)(μX) = the mean of ΣX (√n)(σX) = standard deviation of ΣX To find probabilities for sums on the calculator, follow these steps 2nd DISTR 2:normalcdf normalcdf(lower value of the area, upper value of the area, (n)(mean), (√n)(standard deviation)) 1/10 The Central Limit Theorem for Sums where: • mean is the mean of the original distribution • standard deviation is the standard deviation of the original distribution • sample size = n An unknown distribution has a mean of 90 and a standard deviation of 15 A sample of size 80 is drawn randomly from the population Find the probability that the sum of the 80 values (or the total of the 80 values) is more than 7,500 Find the sum that is 1.5 standard deviations above the mean of the sums Let X = one value from the original unknown population The probability question asks you to find a probability for the sum (or total of) 80 values ΣX = the sum or total of 80 values Since μX = 90, σX = 15, and n = 80, ΣX ~ N((80)(90), (√80)(15)) • mean of the sums = (n)(μX) = (80)(90) = 7,200 • standard deviation of the sums = (√n)(σX) = (√80)(15) • sum of 80 values = Σx = 7,500 a Find P(Σx > 7,500) P(Σx > 7,500) = 0.0127 normalcdf(lower value, upper value, mean of sums, stdev of sums) The parameter list is abbreviated(lower, upper, (n)(μX, (√n)(σX)) normalcdf (7500,1E99,(80)(90),(√80)(15)) = 0.0127 Reminder 2/10 The Central Limit Theorem for Sums 1E99 = 1099 Press the EE key for E b Find Σx where z = 1.5 Σx = (n)(μX) + (z)(√n)(σΧ) = (80)(90) + (1.5)(√80)(15) = 7,401.2 Try It An unknown distribution has a mean of 45 and a standard deviation of eight A sample size of 50 is drawn randomly from the population Find the probability that the sum of the 50 values is more than 2,400 0.0040 To find percentiles for sums on the calculator, follow these steps 2nd DIStR 3:invNorm k = invNorm (area to the left of k, (n)(mean), (√n)(standard deviation)) where: • • • • k is the kth percentile mean is the mean of the original distribution standard deviation is the standard deviation of the original distribution sample size = n In a recent study reported Oct 29, 2012 on the Flurry Blog, the mean age of tablet users is 34 years Suppose the standard deviation is 15 years The sample of size is 50 What are the mean and standard deviation for the sum of the ages of tablet users? What is the distribution? Find the probability that the sum of the ages is between 1,500 and 1,800 years Find the 80th percentile for the sum of the 50 ages μΣx = nμx = 50(34) = 1,700 and σΣx = √nσx = (√50 )(15) = 106.01 The distribution is normal for sums by the central limit theorem P(1500 < Σx < 1800) = normalcdf (1,500, 1,800, (50)(34), (√50 )(15)) = 0.7974 Let k = the 80th percentile k = invNorm(0.80,(50)(34),(√50 )(15)) = 1,789.3 3/10 The Central Limit Theorem for Sums Try It In a recent study reported Oct.29, 2012 on the Flurry Blog, the mean age of tablet users is 35 years Suppose the standard deviation is ten years The sample size is 39 What are the mean and standard deviation for the sum of the ages of tablet users? What is the distribution? Find the probability that the sum of the ages is between 1,400 and 1,500 years Find the 90th percentile for the sum of the 39 ages μΣx = nμx = 1,365 and σΣx = √nσx = 62.4 The distribution is normal for sums by the central limit theorem P(1400 < Σx < 1500) = normalcdf (1400,1500,(39)(35),(√39)(10)) = 0.2723 Let k = the 90th percentile k = invNorm(0.90,(39)(35),(√39) (10)) = 1445.0 The mean number of minutes for app engagement by a tablet user is 8.2 minutes Suppose the standard deviation is one minute Take a sample of size 70 What are the mean and standard deviation for the sums? Find the 95th percentile for the sum of the sample Interpret this value in a complete sentence Find the probability that the sum of the sample is at least ten hours μΣx = nμx = 70(8.2) = 574 minutes and σΣx = (√n)(σx) = (√70 )(1) = 8.37 minutes Let k = the 95th percentile k = invNorm (0.95,(70)(8.2),(√70)(1)) = 587.76 minutes ...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 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 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 ... Review The Central Limit Theorem for Sums: ∑X ~ N[(n)(μx),(√n)(σx)] Mean for Sums (∑X): (n)(μx) The Central Limit Theorem for Sums z-score and standard deviation for sums: zfor the sample mean =... randomly from the population Find the probability that the sum of the 95 values is greater than 7,650 0.3345 5/10 The Central Limit Theorem for Sums Find the probability that the sum of the 95 values... below the mean of the sums Find the percentage of sums between 1.5 standard deviations below the mean of the sums and one standard deviation above the mean of the sums 77.45% Use the following information