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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 Using the Central Limit Theorem Using the Central Limit Theorem By: OpenStaxCollege It is important for you to understand when to use the central limit theorem If you are being asked to find the probability of the mean, use the clt for the mean If you are being asked to find the probability of a sum or total, use the clt for sums This also applies to percentiles for means and sums NOTE If you are being asked to find the probability of an individual value, not use the clt Use the distribution of its random variable Examples of the Central Limit Theorem Law of Large Numbers The law of large numbers says that if you take samples of larger and larger size from ¯ any population, then the mean x of the sample tends to get closer and closer to μ From the central limit theorem, we know that as n gets larger and larger, the sample means follow a normal distribution The larger n gets, the smaller the standard deviation gets ¯ σ ¯ (Remember that the standard deviation for X is √n ) This means that the sample mean x must be close to the population mean μ We can say that μ is the value that the sample means approach as n gets larger The central limit theorem illustrates the law of large numbers Central Limit Theorem for the Mean and Sum Examples A study involving stress is conducted among the students on a college campus The stress scores follow a uniform distribution with the lowest stress score equal to one and the highest equal to five Using a sample of 75 students, find: The probability that the mean stress score for the 75 students is less than two 1/24 Using the Central Limit Theorem The 90th percentile for the mean stress score for the 75 students The probability that the total of the 75 stress scores is less than 200 The 90th percentile for the total stress score for the 75 students Let X = one stress score Problems a and b ask you to find a probability or a percentile for a mean Problems c and d ask you to find a probability or a percentile for a total or sum The sample size, n, is equal to 75 Since the individual stress scores follow a uniform distribution, X ~ U(1, 5) where a = and b = (See Continuous Random Variables for an explanation on the uniform distribution) μX = a+b σX = √ = 1+5 (b – a)2 12 = √ =3 (5 – 1)2 12 = 1.15 ¯ For problems and 2., let X = the mean stress score for the 75 students Then, ¯ X ∼ N 3, ( 1.15 √75 ) where n = 75 ¯ a Find P(x < 2) Draw the graph ¯ a P(x < 2) = The probability that the mean stress score is less than two is about zero ( 1.15 ) normalcdf 1,2,3, √75 = 2/24 Using the Central Limit Theorem Reminder The smallest stress score is one b Find the 90th percentile for the mean of 75 stress scores Draw a graph b Let k = the 90th precentile ¯ Find k, where P(x < k) = 0.90 k = 3.2 The 90th percentile for the mean of 75 scores is about 3.2 This tells us that 90% of all the means of 75 stress scores are at most 3.2, and that 10% are at least 3.2 ( 1.15 ) invNorm 0.90,3, √75 = 3.2 For problems c and d, let ΣX = the sum of the 75 stress scores Then, ΣX ~ N[(75)(3), (√75)(1.15)] c Find P(Σx < 200) Draw the graph c The mean of the sum of 75 stress scores is (75)(3) = 225 The standard deviation of the sum of 75 stress scores is (√75)(1.15) = 9.96 P(Σx < 200) = The probability that the total of 75 scores is less than 200 is about zero 3/24 Using the Central Limit Theorem normalcdf (75,200,(75)(3),(√75)(1.15)) Reminder The smallest total of 75 stress scores is 75, because the smallest single score is one d Find the 90th percentile for the total of 75 stress scores Draw a graph d Let k = the 90th percentile Find k where P(Σx < k) = 0.90 k = 237.8 The 90th percentile for the sum of 75 scores is about 237.8 This tells us that 90% of all the sums of 75 scores are no more than 237.8 and 10% are no less than 237.8 invNorm(0.90,(75)(3),(√75)(1.15)) = 237.8 Try It Use the information in [link], but use a sample size of 55 to answer the following questions ¯ Find P(x < 7) Find P(Σx > 170) Find the 80th percentile for the mean of 55 scores Find the 85th percentile for the sum of 55 scores Solutions 0.0265 0.2789 3.13 173.84 4/24 Using the Central Limit Theorem Suppose that a market research analyst for a cell phone company conducts a study of their customers who exceed the time allowance included on their basic cell phone contract; the analyst finds that for those people who exceed the time included in their basic contract, the excess time used follows an exponential distribution with a mean of 22 minutes Consider a random sample of 80 customers who exceed the time allowance included in their basic cell phone contract Let X = the excess time used by one INDIVIDUAL cell phone customer who exceeds his contracted time allowance X ∼ Exp ( 221 ) From previous chapters, we know that μ = 22 and σ = 22 ¯ Let X = the mean excess time used by a sample of n = 80 customers who exceed their contracted time allowance ¯ X ~ N 22, ( 22 √80 ) by the central limit theorem for sample means Using the clt to find ...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 On the Berry-Esseen bound for a combinatorial
central limit theorem
Th`anh Lˆe Vˇan∗
Abstract
The main finding of this note is an improvement of the Chen-Goldstein-Shao proof of the
Berry-Esseen bound for the combinatorial central limit theorem. A bound of the correct order
in terms of third-moment type quantities with a small explicit constant is obtained. Moreover,
our approach does not need to use a truncation step as in Chen-Goldstein-Shao. An example is
also given to illustrate the optimality of the bound.
Key Words and Phrases: Berry-Esseen bound, combinatorial central limit theorem, zero-bias
coupling, Stein’s method.
2010 Mathematics Subject Classifications: 60F05, 60D05.
1
Introduction and result
Let n ≥ 2 and A = {aij , 1 ≤ i, j ≤ n} be an array of real numbers. In this note, we study the
combinatorial central limit theorem, that is, the central limit theorem for random variables of the
form
n
aiπ(i) ,
Y = YA =
(1.1)
i=1
where π is a random permutation with the uniform distribution over the symmetric group of all
permutations of {1, . . . , n}.
The central limit theorem for YA were proved by Wald and Wolfowitz [17] when the factorization
aij = bi cj holds, and by Hoeffding [11] for general arrays. Bounds on the error in the normal
approximation were later considered by a number of authors. Ho and Chen [10] used a concentration
inequality approach and Stein’s method for exchangeable pairs [16], which yield the optimal rate
only under condition that supij |aij | ≤ C. Bolthausen [1] also used Stein’s method with an inductive
approach, which obtained a bound of the correct order in terms of third-moment type quantities,
but with an unspecified constant. Goldstein [7] employing the zero bias version of Stein’s method
obtained bounds with an explicit constant, but in terms of supi,j |ai,j |. Recently, Chen, Goldstein
∗ Department of Mathematics, Vinh University, Nghe An 42118, Vietnam. A part of research of the second author
is also supported by the Vietnam Institute for Advanced Study in Mathematics (VIASM) and the Vietnam National
Foundation of Sciences and Technology Development (NAFOSTED). Email: levt@vinhuni.edu.vn
1
and Shao [5, Theorem 6.2] used the zero bias variation of Ghosh [6] on the inductive method in
Bolthausen [1] to prove a bound depending on a third moment type quantity of the matrix, but
with an unspecified constant like Bolthausen [1]. In this note, we give an improvement of the
Chen-Goldstein-Shao proof and obtain a bound of the correct order in terms of third-moment type
quantities with a constant c = 90. Moreover, our approach do not need to use the truncation step
as in [1, 5, 6]. We also give an example to illustrate the optimality of the bound.
As far as we are aware, on bound depending on a third moment type quantity of the matrix,
the best absolute constant is c = 447 which was obtained very recently by Chen and Fang [4].
(Neammanee and Suntornchost [15] obtained a constant c = 198. However, Chen and Fang [4]
showed that the proof in [15] is incorrect.) Both in [4] and [15], the authors used the concentration
inequality approach and method of exchangeable pairs, and considered the case where the elements
of A are independent random variables.
2
We denote the mean and variance of YA by µA and σA
, and use the following notation.
ai. =
1
n
n
aij , 1 ≤ i ≤ n, a.j =
j=1
1
n
It is known that
n
aij , 1 ≤ j ≤ n, and a.. =
i=1
n
ai. =
i=1
aij .
i,j=1
a.π(i) ,
(1.2)
i=1
n
2
=
σA
n
n
µA = na.. =
and
1
n2
n
1
1
(a2 − a2i. − a2.j + a2.. ) =
(aij − ai. − a.j + a.. )2 .
n − 1 i,j=1 ij
n − 1 i,j=1
(1.3)
From (1.2), we automatically get that
n
(aiπ(i) − ai. − a.π(i) + a.. ).
YA − EYA =
(1.4)
i=1
We also denote WA = (YA − µA )/σA and
βA =
n
i,j=1
|aij − ai. − a.j + a.. |3
3
σA
.
(1.5)
1
x
Throughout this note, Z is the standard normal random variable, Φ(x) = √
exp(−t2 /2)dt
−∞
2π
is the distribution function of Z. For n ≥ 1, let Sn denote the symmetric group of all 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 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 The Central Limit Theorem for Sample Means (Averages) The Central Limit Theorem for Sample Means (Averages) By: OpenStaxCollege Suppose X is a random variable with a distribution that may be known or unknown (it can be any .. .Using the Central Limit Theorem The 90th percentile for the mean stress score for the 75 students The probability that the total of the 75 stress scores is less than 200 The 90th percentile... 12/24 Using the Central Limit Theorem Draw the graph from [link] Find the probability that the mean actual weight for the 100 weights is greater than 25.2 0.0003 Draw the graph from [link] Find the. .. Review The central limit theorem can be used to illustrate the law of large numbers The law of large numbers states that the larger the sample size you take from a population, the ¯ closer the sample
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