An example of non-quenched convergence in the conditional central limit theorem for partial sums of a linear process
|
|
- Donald Page
- 7 years ago
- Views:
Transcription
1 A example of o-queched covergece i the coditioal cetral limit theorem for partial sums of a liear process Dalibor Volý ad Michael Woodroofe Abstract A causal liear processes X,X 0,X is costructed for which the coditioal distributios of stadardized partial sums S = X + + X give X,X 0 coverge i probability to the stadard ormal distributio, but do ot coverge w.p.. Itroductio Let X,X 0,X, deote a strictly statioary sequece defied o a probability space (Ω,A,) ad adapted to a filtratio F k. Suppose that the X k have mea 0 ad fiite variace; ad let S = X + +X ad σ 2 = E(S 2 ). With this otatio the Coditioal Cetral Limit Questio may be stated: Whe do the coditioal distributios of a S /σ coverge i probability to the stadard ormal distributio; that is, whe does the Levy distace betwee the coditioal distributio ad the stadard ormal coverge i probability zero? Two sets of ecessary ad sufficiet coditios may be foud i 3 ad 0. Oe ca also ask: Whe is the covergece queched; that is, whe do the coditioal distributios coverge almost surely? I a Markov Chai settig, this meas that the covergece takes place for almost every (with respect to the statioary measure) startig poit. It has bee show that several importat classical limit theorems are queched. See ad (for more recet research, e.g.) 5, 2, ad. Dalibor Volý Dalibor.Voly@uiv-roue.fr Michael Woodroofe
2 2 D. Volý ad M. Woodroofe For a causal liear process X k = i=0 a i ξ k i, () where a 0,a, are square summable ad ξ,ξ 0,ξ, are i.i.d. with mea 0 ad variace oe, let F = σ{,ξ,ξ }. The S = i=0 b i+ b i ξ i + i= b i ξ i, where b = a a. It follows that E(S F 0 ) = i=0 b i+ b i ξ i ad S E(S F 0 ) = i= b iξ i are idepedet. The, lettig deote the orm i L 2 () ad τ 2 = b b2, ad E(S F 0 ) 2 = i=0 b i+ b i 2 = ν 2 say, σ 2 = EE(S F 0 ) 2 +E{S E(S F 0 ) 2 } = ν 2 + τ2. With this otatio, Wu ad Woodroofe 0 showed that if lim σ 2 =, the the coditioal distributio fuctio of S /σ give F 0 coverges i probability to the stadard ormal distributio iff ν 2 = o(σ2 ). Here a causal liear process with absolutely summable coefficiets is costructed for which the coditioal distributios of S /σ coverge to Φ i probability but ot with probability oe. The summability of coefficiets meas that Haa s coditio 7, i 0 E(X 0 F i ) E(X 0 F i ) <, is satisfied ad, therefore, that the (weak) ivariace priciple holds (see 4). 2 The prelimiaries The first step is to develop a ecessary ad sufficiet coditio for queched covergece. The proof of the lemma below uses the Covergece of Types Theorem (8, p. 203): Let Y ad Z be radom variables of the form Y = α Z + β, where α, β R are costats. If Y Y ad Z Z, where Z o-degeerate, the α ad β coverge to limits α ad β ad Y = dist αz + β. Lemma For a causal liear process () for which σ ad ν = o(σ ), the coditioal distributio of S /σ give F 0 coverges to the stadard ormal distributio w.p. iff lim E(S F 0 ) = 0 w.p.. (2) σ roof. Suppose first that ν = o(σ ) i which case τ 2 /σ2, ad let F deote the (ucoditioal) distributio fuctio of S E(S F 0 )/τ. The S σ z E(S z F F 0 ) 0 = F, σ τ
3 No-queched covergece i the coditioal cetral limit theorem 3 by idepedece. It is show below that F coverges weakly to the stadard ormal distributio Φ. The sufficiecy of (2) for almost sure covergece of the coditioal distributio fuctio is the obvious. Coversely, if the coditioal distributios coverge almost surely to the stadard ormal distributio, the E(S F 0 )/σ 0 w.p., by the Covergece of Types Theorem, applied coditioally. That F Φ follows from Theorem 2. ad Corollary 2. of 9. The argumet is sketched here for completeess. By the Lideberg Feller Coditio, it is sufficiet to show that lim b 2 i 0 i τ 2 = 0. (3) Suppose that the imum is attaied at i ad (temporarily) that i 2; ad let A 2 = i=0 a2 i. If k 2, the b 2 i b 2 i +k = k k i= (b i + j b i + j)(b i + j + b i + j) a 2 i + j k (b i + j + b i + j) 2 2Aτ. So, for ay m 2, mb2 i m k= b2 i +k + 2Amτ τ 2 + 2Amτ ad, therefore b 2 i τ 2 m + 2A τ. The same iequality may be obtaied if i 2 by a dual argumet i which k is replaced by k, ad (3) follows by lettig ad m i that order. Lemma 2 For a causal liear process (): If a 0,a,a 2, are absolutely summable ad b := i=0 a i 0, the σ 2 b 2 ad ν 2 = o(σ 2 ). roof. The proof uses a differet expressio for E(S F 0 ), E(S F 0 ) = a j ζ j + a j ζ j ζ j, j=+ where ζ j = ξ j+ + + ξ 0. Thus, τ 2 = b b2 b2 ad E(S F 0 ) a j j + a j = o( ) = o(σ ). j=+ The lemma follows directly.
4 4 D. Volý ad M. Woodroofe 3 The example The mai result follows. Theorem There are o-egative summable coefficiets a 0,a,a 2, for which ν 2 = o(σ 2 ) but (2) fails. roof. By Lemma 2, it suffices to costruct positive summable coefficiets a 0,a,a 2, for which (2) fails. We cosider coefficiets of the form a j = 2 j j, (4) for j, ad a i = 0 if i / {, 2, }, where 0 = 0 < < 2 < is a sequece of positive itegers costructed below. It is clear that a sequece of the form (4) is summable. For sequeces of the form (4), E(S F 0 ) = j a j ζ j + j > a j ζ j ζ j, where ζ = ξ ξ 0, as above. So, for k < k where E(S F 0 ) = I k ()+II k (), k I k () = II k () = a j ζ j + a k ζ k ζ k, (5) a j ζ j ζ j, j=k+ ad the empty sum is to be iterpreted as zero whe k =. The specificatio of the itegers i depeds o the followig claim: There are itegers 0 < < 2 < for which I k () k < k > 2 k+ > ( ) k+ (6) 2 for all k. The proof of the claim, i tur, depeds o the followig observatio: Let J(N,) = ζ N ζ N = ξ N+ + + ξ N+ for N, so that I k () = k a j ζ j +a k J( k,) for k. The the joit distributio of J(N,), N, is the same as the joit distributio of ζ = ξ + + ξ, N. It the follows from the Law of the Iterated Logarithm that 0 <N J(N, ) > 8 = 0 <N ζ > 8
5 No-queched covergece i the coditioal cetral limit theorem 5 as N. So the right side is at least 3/4 all sufficietly large N, ad the existece of follows (sice I ()=J(,)/2). Now, let k 2 ad suppose that,, k have bee costructed. The λ k ca be chose so that k ( ) k+2 a j ζ j > λ k. 2 As above, k <N = J(N, ) > 2 k k (λ k + 2 k+ ) k <N ζ > 2 k k (λ k + 2 k+ ), which approaches as N by the Law of the Iterated Logarithm. The existece of k i (6) follows directly from the last two displays ad (5). The existece of the sequece the follows from mathematical iductio (the existece of,, k implies that of k ). The ext step is to boud the term II k (). By Doob s (953) imal iequality, E k ζ k 2 for all. The So So, E II k () k < k a j E j=k+ j=k+ II k () > 2 k k < k ζ j ζ j k < k 2 k 2 j j 2 k. 2 2k. That (2) fails for this costructio may be see as follows: Clearly, E(S F 0 ) k < k k < k I k () II k (). k < k k < k k E(S F 0 ) 2 k I k () 2 k+ k < k + II k () 2 k k < k for k 2. It follows that E(S F 0 ) 2, k ifiitely ofte = 0 k < k 2 2 k ad, therefore, that limsup E(S F 0 )/ = w.p..
6 6 D. Volý ad M. Woodroofe Refereces A. N. Borodi ad I. A. Ibragimov, Limit theorems for fuctioals of radom walks, Trudy Mat. Ist. Steklov. 95 (994). Trasl. ito Eglish: roc. Steklov Ist. Math. 95 (995), o.2. 2 C. Cuy, oitwise ergodic theorems with rate ad applicatio to limit theorems for statioary processes, arxiv: v, J. Dedecker ad F. Merlevède, Necessary ad sufficiet coditios for the coditioal cetral limit theorem, A. rob. 32 (2002), J. Dedecker, F. Merlevède, ad D. Volý, O the weak ivariace priciple for o adapted sequeces uder projective criteria, Joural of Theoretical robability 20 (2007), Y. Derrieic ad M. Li, The cetral limit theorem for Markov chais with ormal trasitio operators started at a poit, robab. Theory Relat. Fields 9 (200), J. Doob, Stochastic rocesses, Wiley, E. J. Haa, Cetral limit theorems for time series regressio, Z. Wahrscheilichkeitstheorie verw. Geb. 26 (973), M. Loeve, robability Theory, Va Nostrad, M. eligrad ad S. Utev, Cetral limit theorem for liear processes, A. rob. 25 (997), Wei-Biao Wu ad M. Woodroofe, Martigale approximatios for sums of statioary processes, A. rob. 32 (2004), Ou Zhao ad M. Woodroofe, Law of the iterated logarithm for statioary processes, A. rob. 37, (2007),
Properties of MLE: consistency, asymptotic normality. Fisher information.
Lecture 3 Properties of MLE: cosistecy, asymptotic ormality. Fisher iformatio. I this sectio we will try to uderstad why MLEs are good. Let us recall two facts from probability that we be used ofte throughout
More informationLecture 4: Cauchy sequences, Bolzano-Weierstrass, and the Squeeze theorem
Lecture 4: Cauchy sequeces, Bolzao-Weierstrass, ad the Squeeze theorem The purpose of this lecture is more modest tha the previous oes. It is to state certai coditios uder which we are guarateed that limits
More informationOverview of some probability distributions.
Lecture Overview of some probability distributios. I this lecture we will review several commo distributios that will be used ofte throughtout the class. Each distributio is usually described by its probability
More informationSAMPLE QUESTIONS FOR FINAL EXAM. (1) (2) (3) (4) Find the following using the definition of the Riemann integral: (2x + 1)dx
SAMPLE QUESTIONS FOR FINAL EXAM REAL ANALYSIS I FALL 006 3 4 Fid the followig usig the defiitio of the Riema itegral: a 0 x + dx 3 Cosider the partitio P x 0 3, x 3 +, x 3 +,......, x 3 3 + 3 of the iterval
More informationChapter 7 Methods of Finding Estimators
Chapter 7 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 011 Chapter 7 Methods of Fidig Estimators Sectio 7.1 Itroductio Defiitio 7.1.1 A poit estimator is ay fuctio W( X) W( X1, X,, X ) of
More informationChapter 6: Variance, the law of large numbers and the Monte-Carlo method
Chapter 6: Variace, the law of large umbers ad the Mote-Carlo method Expected value, variace, ad Chebyshev iequality. If X is a radom variable recall that the expected value of X, E[X] is the average value
More informationAsymptotic Growth of Functions
CMPS Itroductio to Aalysis of Algorithms Fall 3 Asymptotic Growth of Fuctios We itroduce several types of asymptotic otatio which are used to compare the performace ad efficiecy of algorithms As we ll
More informationUniversity of California, Los Angeles Department of Statistics. Distributions related to the normal distribution
Uiversity of Califoria, Los Ageles Departmet of Statistics Statistics 100B Istructor: Nicolas Christou Three importat distributios: Distributios related to the ormal distributio Chi-square (χ ) distributio.
More informationPROCEEDINGS OF THE YEREVAN STATE UNIVERSITY AN ALTERNATIVE MODEL FOR BONUS-MALUS SYSTEM
PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY Physical ad Mathematical Scieces 2015, 1, p. 15 19 M a t h e m a t i c s AN ALTERNATIVE MODEL FOR BONUS-MALUS SYSTEM A. G. GULYAN Chair of Actuarial Mathematics
More informationUC Berkeley Department of Electrical Engineering and Computer Science. EE 126: Probablity and Random Processes. Solutions 9 Spring 2006
Exam format UC Bereley Departmet of Electrical Egieerig ad Computer Sciece EE 6: Probablity ad Radom Processes Solutios 9 Sprig 006 The secod midterm will be held o Wedesday May 7; CHECK the fial exam
More informationI. Chi-squared Distributions
1 M 358K Supplemet to Chapter 23: CHI-SQUARED DISTRIBUTIONS, T-DISTRIBUTIONS, AND DEGREES OF FREEDOM To uderstad t-distributios, we first eed to look at aother family of distributios, the chi-squared distributios.
More informationConvexity, Inequalities, and Norms
Covexity, Iequalities, ad Norms Covex Fuctios You are probably familiar with the otio of cocavity of fuctios. Give a twicedifferetiable fuctio ϕ: R R, We say that ϕ is covex (or cocave up) if ϕ (x) 0 for
More informationSampling Distribution And Central Limit Theorem
() Samplig Distributio & Cetral Limit Samplig Distributio Ad Cetral Limit Samplig distributio of the sample mea If we sample a umber of samples (say k samples where k is very large umber) each of size,
More informationIn nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008
I ite Sequeces Dr. Philippe B. Laval Keesaw State Uiversity October 9, 2008 Abstract This had out is a itroductio to i ite sequeces. mai de itios ad presets some elemetary results. It gives the I ite Sequeces
More informationSection 11.3: The Integral Test
Sectio.3: The Itegral Test Most of the series we have looked at have either diverged or have coverged ad we have bee able to fid what they coverge to. I geeral however, the problem is much more difficult
More informationA probabilistic proof of a binomial identity
A probabilistic proof of a biomial idetity Joatho Peterso Abstract We give a elemetary probabilistic proof of a biomial idetity. The proof is obtaied by computig the probability of a certai evet i two
More informationTheorems About Power Series
Physics 6A Witer 20 Theorems About Power Series Cosider a power series, f(x) = a x, () where the a are real coefficiets ad x is a real variable. There exists a real o-egative umber R, called the radius
More informationChapter 7 - Sampling Distributions. 1 Introduction. What is statistics? It consist of three major areas:
Chapter 7 - Samplig Distributios 1 Itroductio What is statistics? It cosist of three major areas: Data Collectio: samplig plas ad experimetal desigs Descriptive Statistics: umerical ad graphical summaries
More informationIrreducible polynomials with consecutive zero coefficients
Irreducible polyomials with cosecutive zero coefficiets Theodoulos Garefalakis Departmet of Mathematics, Uiversity of Crete, 71409 Heraklio, Greece Abstract Let q be a prime power. We cosider the problem
More information0.7 0.6 0.2 0 0 96 96.5 97 97.5 98 98.5 99 99.5 100 100.5 96.5 97 97.5 98 98.5 99 99.5 100 100.5
Sectio 13 Kolmogorov-Smirov test. Suppose that we have a i.i.d. sample X 1,..., X with some ukow distributio P ad we would like to test the hypothesis that P is equal to a particular distributio P 0, i.e.
More informationOur aim is to show that under reasonable assumptions a given 2π-periodic function f can be represented as convergent series
8 Fourier Series Our aim is to show that uder reasoable assumptios a give -periodic fuctio f ca be represeted as coverget series f(x) = a + (a cos x + b si x). (8.) By defiitio, the covergece of the series
More informationTHE ABRACADABRA PROBLEM
THE ABRACADABRA PROBLEM FRANCESCO CARAVENNA Abstract. We preset a detailed solutio of Exercise E0.6 i [Wil9]: i a radom sequece of letters, draw idepedetly ad uiformly from the Eglish alphabet, the expected
More informationSequences and Series
CHAPTER 9 Sequeces ad Series 9.. Covergece: Defiitio ad Examples Sequeces The purpose of this chapter is to itroduce a particular way of geeratig algorithms for fidig the values of fuctios defied by their
More information4.3. The Integral and Comparison Tests
4.3. THE INTEGRAL AND COMPARISON TESTS 9 4.3. The Itegral ad Compariso Tests 4.3.. The Itegral Test. Suppose f is a cotiuous, positive, decreasig fuctio o [, ), ad let a = f(). The the covergece or divergece
More informationHypothesis testing. Null and alternative hypotheses
Hypothesis testig Aother importat use of samplig distributios is to test hypotheses about populatio parameters, e.g. mea, proportio, regressio coefficiets, etc. For example, it is possible to stipulate
More informationConfidence Intervals for One Mean
Chapter 420 Cofidece Itervals for Oe Mea Itroductio This routie calculates the sample size ecessary to achieve a specified distace from the mea to the cofidece limit(s) at a stated cofidece level for a
More informationA Note on Sums of Greatest (Least) Prime Factors
It. J. Cotemp. Math. Scieces, Vol. 8, 203, o. 9, 423-432 HIKARI Ltd, www.m-hikari.com A Note o Sums of Greatest (Least Prime Factors Rafael Jakimczuk Divisio Matemática, Uiversidad Nacioal de Luá Bueos
More informationLecture 5: Span, linear independence, bases, and dimension
Lecture 5: Spa, liear idepedece, bases, ad dimesio Travis Schedler Thurs, Sep 23, 2010 (versio: 9/21 9:55 PM) 1 Motivatio Motivatio To uderstad what it meas that R has dimesio oe, R 2 dimesio 2, etc.;
More informationLECTURE NOTES ON DONSKER S THEOREM
LECTURE NOTES ON DONSKER S THEOREM DAVAR KHOSHNEVISAN ABSTRACT. Some course otes o Dosker s theorem. These are for Math 7880-1 Topics i Probability, taught at the Deparmet of Mathematics at the Uiversity
More informationA note on weak convergence of the sequential multivariate empirical process under strong mixing SFB 823. Discussion Paper.
SFB 823 A ote o weak covergece of the sequetial multivariate empirical process uder strog mixig Discussio Paper Axel Bücher Nr. 17/2013 A ote o weak covergece of the sequetial multivariate empirical process
More information1. MATHEMATICAL INDUCTION
1. MATHEMATICAL INDUCTION EXAMPLE 1: Prove that for ay iteger 1. Proof: 1 + 2 + 3 +... + ( + 1 2 (1.1 STEP 1: For 1 (1.1 is true, sice 1 1(1 + 1. 2 STEP 2: Suppose (1.1 is true for some k 1, that is 1
More informationA RANDOM PERMUTATION MODEL ARISING IN CHEMISTRY
J. Appl. Prob. 45, 060 070 2008 Prited i Eglad Applied Probability Trust 2008 A RANDOM PERMUTATION MODEL ARISING IN CHEMISTRY MARK BROWN, The City College of New York EROL A. PEKÖZ, Bosto Uiversity SHELDON
More informationInfinite Sequences and Series
CHAPTER 4 Ifiite Sequeces ad Series 4.1. Sequeces A sequece is a ifiite ordered list of umbers, for example the sequece of odd positive itegers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29...
More informationLecture 13. Lecturer: Jonathan Kelner Scribe: Jonathan Pines (2009)
18.409 A Algorithmist s Toolkit October 27, 2009 Lecture 13 Lecturer: Joatha Keler Scribe: Joatha Pies (2009) 1 Outlie Last time, we proved the Bru-Mikowski iequality for boxes. Today we ll go over the
More informationTHE LEAST COMMON MULTIPLE OF A QUADRATIC SEQUENCE
THE LEAST COMMON MULTIPLE OF A QUADRATIC SEQUENCE JAVIER CILLERUELO Abstract. We obtai, for ay irreducible quadratic olyomial f(x = ax 2 + bx + c, the asymtotic estimate log l.c.m. {f(1,..., f(} log. Whe
More informationWHEN IS THE (CO)SINE OF A RATIONAL ANGLE EQUAL TO A RATIONAL NUMBER?
WHEN IS THE (CO)SINE OF A RATIONAL ANGLE EQUAL TO A RATIONAL NUMBER? JÖRG JAHNEL 1. My Motivatio Some Sort of a Itroductio Last term I tought Topological Groups at the Göttige Georg August Uiversity. This
More informationTHE HEIGHT OF q-binary SEARCH TREES
THE HEIGHT OF q-binary SEARCH TREES MICHAEL DRMOTA AND HELMUT PRODINGER Abstract. q biary search trees are obtaied from words, equipped with the geometric distributio istead of permutatios. The average
More informationActa Acad. Paed. Agriensis, Sectio Mathematicae 29 (2002) 77 87. ALMOST SURE FUNCTIONAL LIMIT THEOREMS IN L p( ]0, 1[ ), WHERE 1 p <
Acta Acad. Paed. Agriesis, Sectio Mathematicae 29 22) 77 87 ALMOST SUR FUNCTIONAL LIMIT THORMS IN L ], [ ), WHR < József Túri Nyíregyháza, Hugary) Dedicated to the memory of Professor Péter Kiss Abstract.
More informationOutput Analysis (2, Chapters 10 &11 Law)
B. Maddah ENMG 6 Simulatio 05/0/07 Output Aalysis (, Chapters 10 &11 Law) Comparig alterative system cofiguratio Sice the output of a simulatio is radom, the comparig differet systems via simulatio should
More informationCase Study. Normal and t Distributions. Density Plot. Normal Distributions
Case Study Normal ad t Distributios Bret Halo ad Bret Larget Departmet of Statistics Uiversity of Wiscosi Madiso October 11 13, 2011 Case Study Body temperature varies withi idividuals over time (it ca
More informationDepartment of Computer Science, University of Otago
Departmet of Computer Sciece, Uiversity of Otago Techical Report OUCS-2006-09 Permutatios Cotaiig May Patters Authors: M.H. Albert Departmet of Computer Sciece, Uiversity of Otago Micah Colema, Rya Fly
More informationFIBONACCI NUMBERS: AN APPLICATION OF LINEAR ALGEBRA. 1. Powers of a matrix
FIBONACCI NUMBERS: AN APPLICATION OF LINEAR ALGEBRA. Powers of a matrix We begi with a propositio which illustrates the usefuless of the diagoalizatio. Recall that a square matrix A is diogaalizable if
More informationOn the L p -conjecture for locally compact groups
Arch. Math. 89 (2007), 237 242 c 2007 Birkhäuser Verlag Basel/Switzerlad 0003/889X/030237-6, ublished olie 2007-08-0 DOI 0.007/s0003-007-993-x Archiv der Mathematik O the L -cojecture for locally comact
More informationON AN INTEGRAL OPERATOR WHICH PRESERVE THE UNIVALENCE
Proceedigs of the Iteratioal Coferece o Theory ad Applicatios of Mathematics ad Iformatics ICTAMI 3, Alba Iulia ON AN INTEGRAL OPERATOR WHICH PRESERVE THE UNIVALENCE by Maria E Gageoea ad Silvia Moldoveau
More information1 Computing the Standard Deviation of Sample Means
Computig the Stadard Deviatio of Sample Meas Quality cotrol charts are based o sample meas ot o idividual values withi a sample. A sample is a group of items, which are cosidered all together for our aalysis.
More informationARTICLE IN PRESS. Statistics & Probability Letters ( ) A Kolmogorov-type test for monotonicity of regression. Cecile Durot
STAPRO 66 pp: - col.fig.: il ED: MG PROD. TYPE: COM PAGN: Usha.N -- SCAN: il Statistics & Probability Letters 2 2 2 2 Abstract A Kolmogorov-type test for mootoicity of regressio Cecile Durot Laboratoire
More informationON THE DENSE TRAJECTORY OF LASOTA EQUATION
UNIVERSITATIS IAGELLONICAE ACTA MATHEMATICA, FASCICULUS XLIII 2005 ON THE DENSE TRAJECTORY OF LASOTA EQUATION by Atoi Leo Dawidowicz ad Najemedi Haribash Abstract. I preseted paper the dese trajectory
More informationNOTES ON PROBABILITY Greg Lawler Last Updated: March 21, 2016
NOTES ON PROBBILITY Greg Lawler Last Updated: March 21, 2016 Overview This is a itroductio to the mathematical foudatios of probability theory. It is iteded as a supplemet or follow-up to a graduate course
More informationAnnuities Under Random Rates of Interest II By Abraham Zaks. Technion I.I.T. Haifa ISRAEL and Haifa University Haifa ISRAEL.
Auities Uder Radom Rates of Iterest II By Abraham Zas Techio I.I.T. Haifa ISRAEL ad Haifa Uiversity Haifa ISRAEL Departmet of Mathematics, Techio - Israel Istitute of Techology, 3000, Haifa, Israel I memory
More informationMARTINGALES AND A BASIC APPLICATION
MARTINGALES AND A BASIC APPLICATION TURNER SMITH Abstract. This paper will develop the measure-theoretic approach to probability i order to preset the defiitio of martigales. From there we will apply this
More informationTHE REGRESSION MODEL IN MATRIX FORM. For simple linear regression, meaning one predictor, the model is. for i = 1, 2, 3,, n
We will cosider the liear regressio model i matrix form. For simple liear regressio, meaig oe predictor, the model is i = + x i + ε i for i =,,,, This model icludes the assumptio that the ε i s are a sample
More informationThe following example will help us understand The Sampling Distribution of the Mean. C1 C2 C3 C4 C5 50 miles 84 miles 38 miles 120 miles 48 miles
The followig eample will help us uderstad The Samplig Distributio of the Mea Review: The populatio is the etire collectio of all idividuals or objects of iterest The sample is the portio of the populatio
More informationLecture 4: Cheeger s Inequality
Spectral Graph Theory ad Applicatios WS 0/0 Lecture 4: Cheeger s Iequality Lecturer: Thomas Sauerwald & He Su Statemet of Cheeger s Iequality I this lecture we assume for simplicity that G is a d-regular
More informationMeasures of Spread and Boxplots Discrete Math, Section 9.4
Measures of Spread ad Boxplots Discrete Math, Sectio 9.4 We start with a example: Example 1: Comparig Mea ad Media Compute the mea ad media of each data set: S 1 = {4, 6, 8, 10, 1, 14, 16} S = {4, 7, 9,
More information3 Basic Definitions of Probability Theory
3 Basic Defiitios of Probability Theory 3defprob.tex: Feb 10, 2003 Classical probability Frequecy probability axiomatic probability Historical developemet: Classical Frequecy Axiomatic The Axiomatic defiitio
More informationTHIN SEQUENCES AND THE GRAM MATRIX PAMELA GORKIN, JOHN E. MCCARTHY, SANDRA POTT, AND BRETT D. WICK
THIN SEQUENCES AND THE GRAM MATRIX PAMELA GORKIN, JOHN E MCCARTHY, SANDRA POTT, AND BRETT D WICK Abstract We provide a ew proof of Volberg s Theorem characterizig thi iterpolatig sequeces as those for
More informationMaximum Likelihood Estimators.
Lecture 2 Maximum Likelihood Estimators. Matlab example. As a motivatio, let us look at oe Matlab example. Let us geerate a radom sample of size 00 from beta distributio Beta(5, 2). We will lear the defiitio
More informationBASIC STATISTICS. f(x 1,x 2,..., x n )=f(x 1 )f(x 2 ) f(x n )= f(x i ) (1)
BASIC STATISTICS. SAMPLES, RANDOM SAMPLING AND SAMPLE STATISTICS.. Radom Sample. The radom variables X,X 2,..., X are called a radom sample of size from the populatio f(x if X,X 2,..., X are mutually idepedet
More information3. Greatest Common Divisor - Least Common Multiple
3 Greatest Commo Divisor - Least Commo Multiple Defiitio 31: The greatest commo divisor of two atural umbers a ad b is the largest atural umber c which divides both a ad b We deote the greatest commo gcd
More information1 The Gaussian channel
ECE 77 Lecture 0 The Gaussia chael Objective: I this lecture we will lear about commuicatio over a chael of practical iterest, i which the trasmitted sigal is subjected to additive white Gaussia oise.
More informationA Recursive Formula for Moments of a Binomial Distribution
A Recursive Formula for Momets of a Biomial Distributio Árpád Béyi beyi@mathumassedu, Uiversity of Massachusetts, Amherst, MA 01003 ad Saverio M Maago smmaago@psavymil Naval Postgraduate School, Moterey,
More informationChapter 14 Nonparametric Statistics
Chapter 14 Noparametric Statistics A.K.A. distributio-free statistics! Does ot deped o the populatio fittig ay particular type of distributio (e.g, ormal). Sice these methods make fewer assumptios, they
More informationAMS 2000 subject classification. Primary 62G08, 62G20; secondary 62G99
VARIABLE SELECTION IN NONPARAMETRIC ADDITIVE MODELS Jia Huag 1, Joel L. Horowitz 2 ad Fegrog Wei 3 1 Uiversity of Iowa, 2 Northwester Uiversity ad 3 Uiversity of West Georgia Abstract We cosider a oparametric
More informationApproximating Area under a curve with rectangles. To find the area under a curve we approximate the area using rectangles and then use limits to find
1.8 Approximatig Area uder a curve with rectagles 1.6 To fid the area uder a curve we approximate the area usig rectagles ad the use limits to fid 1.4 the area. Example 1 Suppose we wat to estimate 1.
More informationDiscrete Mathematics and Probability Theory Spring 2014 Anant Sahai Note 13
EECS 70 Discrete Mathematics ad Probability Theory Sprig 2014 Aat Sahai Note 13 Itroductio At this poit, we have see eough examples that it is worth just takig stock of our model of probability ad may
More informationMath C067 Sampling Distributions
Math C067 Samplig Distributios Sample Mea ad Sample Proportio Richard Beigel Some time betwee April 16, 2007 ad April 16, 2007 Examples of Samplig A pollster may try to estimate the proportio of voters
More information10-705/36-705 Intermediate Statistics
0-705/36-705 Itermediate Statistics Larry Wasserma http://www.stat.cmu.edu/~larry/=stat705/ Fall 0 Week Class I Class II Day III Class IV Syllabus August 9 Review Review, Iequalities Iequalities September
More informationChapter 11 Convergence in Distribution
Chapter Covergece i Distributio. Weak covergece i metric spaces 2. Weak covergece i R 3. Tightess ad subsequeces 4. Metrizig weak covergece 5. Characterizig weak covergece i spaces of fuctios 2 Chapter
More information5: Introduction to Estimation
5: Itroductio to Estimatio Cotets Acroyms ad symbols... 1 Statistical iferece... Estimatig µ with cofidece... 3 Samplig distributio of the mea... 3 Cofidece Iterval for μ whe σ is kow before had... 4 Sample
More informationParameter estimation for nonlinear models: Numerical approaches to solving the inverse problem. Lecture 11 04/01/2008. Sven Zenker
Parameter estimatio for oliear models: Numerical approaches to solvig the iverse problem Lecture 11 04/01/2008 Sve Zeker Review: Trasformatio of radom variables Cosider probability distributio of a radom
More informationhp calculators HP 12C Statistics - average and standard deviation Average and standard deviation concepts HP12C average and standard deviation
HP 1C Statistics - average ad stadard deviatio Average ad stadard deviatio cocepts HP1C average ad stadard deviatio Practice calculatig averages ad stadard deviatios with oe or two variables HP 1C Statistics
More informationRunning Time ( 3.1) Analysis of Algorithms. Experimental Studies ( 3.1.1) Limitations of Experiments. Pseudocode ( 3.1.2) Theoretical Analysis
Ruig Time ( 3.) Aalysis of Algorithms Iput Algorithm Output A algorithm is a step-by-step procedure for solvig a problem i a fiite amout of time. Most algorithms trasform iput objects ito output objects.
More informationTaking DCOP to the Real World: Efficient Complete Solutions for Distributed Multi-Event Scheduling
Taig DCOP to the Real World: Efficiet Complete Solutios for Distributed Multi-Evet Schedulig Rajiv T. Maheswara, Milid Tambe, Emma Bowrig, Joatha P. Pearce, ad Pradeep araatham Uiversity of Souther Califoria
More informationThe Stable Marriage Problem
The Stable Marriage Problem William Hut Lae Departmet of Computer Sciece ad Electrical Egieerig, West Virgiia Uiversity, Morgatow, WV William.Hut@mail.wvu.edu 1 Itroductio Imagie you are a matchmaker,
More informationClass Meeting # 16: The Fourier Transform on R n
MATH 18.152 COUSE NOTES - CLASS MEETING # 16 18.152 Itroductio to PDEs, Fall 2011 Professor: Jared Speck Class Meetig # 16: The Fourier Trasform o 1. Itroductio to the Fourier Trasform Earlier i the course,
More information5 Boolean Decision Trees (February 11)
5 Boolea Decisio Trees (February 11) 5.1 Graph Coectivity Suppose we are give a udirected graph G, represeted as a boolea adjacecy matrix = (a ij ), where a ij = 1 if ad oly if vertices i ad j are coected
More informationChapter 7: Confidence Interval and Sample Size
Chapter 7: Cofidece Iterval ad Sample Size Learig Objectives Upo successful completio of Chapter 7, you will be able to: Fid the cofidece iterval for the mea, proportio, ad variace. Determie the miimum
More informationNormal Distribution.
Normal Distributio www.icrf.l Normal distributio I probability theory, the ormal or Gaussia distributio, is a cotiuous probability distributio that is ofte used as a first approimatio to describe realvalued
More informationBasic Elements of Arithmetic Sequences and Series
MA40S PRE-CALCULUS UNIT G GEOMETRIC SEQUENCES CLASS NOTES (COMPLETED NO NEED TO COPY NOTES FROM OVERHEAD) Basic Elemets of Arithmetic Sequeces ad Series Objective: To establish basic elemets of arithmetic
More informationDescriptive Statistics
Descriptive Statistics We leared to describe data sets graphically. We ca also describe a data set umerically. Measures of Locatio Defiitio The sample mea is the arithmetic average of values. We deote
More informationA Test of Normality. 1 n S 2 3. n 1. Now introduce two new statistics. The sample skewness is defined as:
A Test of Normality Textbook Referece: Chapter. (eighth editio, pages 59 ; seveth editio, pages 6 6). The calculatio of p values for hypothesis testig typically is based o the assumptio that the populatio
More informationINFINITE SERIES KEITH CONRAD
INFINITE SERIES KEITH CONRAD. Itroductio The two basic cocepts of calculus, differetiatio ad itegratio, are defied i terms of limits (Newto quotiets ad Riema sums). I additio to these is a third fudametal
More informationIncremental calculation of weighted mean and variance
Icremetal calculatio of weighted mea ad variace Toy Fich faf@cam.ac.uk dot@dotat.at Uiversity of Cambridge Computig Service February 009 Abstract I these otes I eplai how to derive formulae for umerically
More information, a Wishart distribution with n -1 degrees of freedom and scale matrix.
UMEÅ UNIVERSITET Matematisk-statistiska istitutioe Multivariat dataaalys D MSTD79 PA TENTAMEN 004-0-9 LÖSNINGSFÖRSLAG TILL TENTAMEN I MATEMATISK STATISTIK Multivariat dataaalys D, 5 poäg.. Assume that
More informationDegree of Approximation of Continuous Functions by (E, q) (C, δ) Means
Ge. Math. Notes, Vol. 11, No. 2, August 2012, pp. 12-19 ISSN 2219-7184; Copyright ICSRS Publicatio, 2012 www.i-csrs.org Available free olie at http://www.gema.i Degree of Approximatio of Cotiuous Fuctios
More informationVirtile Reguli And Radiational Optaprints
RANDOM GRAPHS WITH FORBIDDEN VERTEX DEGREES GEOFFREY GRIMMETT AND SVANTE JANSON Abstract. We study the radom graph G,λ/ coditioed o the evet that all vertex degrees lie i some give subset S of the oegative
More informationCHAPTER 7: Central Limit Theorem: CLT for Averages (Means)
CHAPTER 7: Cetral Limit Theorem: CLT for Averages (Meas) X = the umber obtaied whe rollig oe six sided die oce. If we roll a six sided die oce, the mea of the probability distributio is X P(X = x) Simulatio:
More informationA Constant-Factor Approximation Algorithm for the Link Building Problem
A Costat-Factor Approximatio Algorithm for the Lik Buildig Problem Marti Olse 1, Aastasios Viglas 2, ad Ilia Zvedeiouk 2 1 Ceter for Iovatio ad Busiess Developmet, Istitute of Busiess ad Techology, Aarhus
More informationLecture 3. denote the orthogonal complement of S k. Then. 1 x S k. n. 2 x T Ax = ( ) λ x. with x = 1, we have. i = λ k x 2 = λ k.
18.409 A Algorithmist s Toolkit September 17, 009 Lecture 3 Lecturer: Joatha Keler Scribe: Adre Wibisoo 1 Outlie Today s lecture covers three mai parts: Courat-Fischer formula ad Rayleigh quotiets The
More informationInference on Proportion. Chapter 8 Tests of Statistical Hypotheses. Sampling Distribution of Sample Proportion. Confidence Interval
Chapter 8 Tests of Statistical Hypotheses 8. Tests about Proportios HT - Iferece o Proportio Parameter: Populatio Proportio p (or π) (Percetage of people has o health isurace) x Statistic: Sample Proportio
More informationA Mathematical Perspective on Gambling
A Mathematical Perspective o Gamblig Molly Maxwell Abstract. This paper presets some basic topics i probability ad statistics, icludig sample spaces, probabilistic evets, expectatios, the biomial ad ormal
More informationDoktori értekezés Katona Zsolt 2006
Doktori értekezés Katoa Zsolt 2006 Radom Graph Models Doktori értekezés Iformatika Doktori Iskola, Az Iformatika Alapjai program, vezető: Demetrovics Jáos Katoa Zsolt Témavezető: Móri Tamás, doces Eötvös
More informationPractice Problems for Test 3
Practice Problems for Test 3 Note: these problems oly cover CIs ad hypothesis testig You are also resposible for kowig the samplig distributio of the sample meas, ad the Cetral Limit Theorem Review all
More informationTHE TWO-VARIABLE LINEAR REGRESSION MODEL
THE TWO-VARIABLE LINEAR REGRESSION MODEL Herma J. Bieres Pesylvaia State Uiversity April 30, 202. Itroductio Suppose you are a ecoomics or busiess maor i a college close to the beach i the souther part
More informationarxiv:0903.5136v2 [math.pr] 13 Oct 2009
First passage percolatio o rado graphs with fiite ea degrees Shakar Bhaidi Reco va der Hofstad Gerard Hooghiestra October 3, 2009 arxiv:0903.536v2 [ath.pr 3 Oct 2009 Abstract We study first passage percolatio
More informationS. Tanny MAT 344 Spring 1999. be the minimum number of moves required.
S. Tay MAT 344 Sprig 999 Recurrece Relatios Tower of Haoi Let T be the miimum umber of moves required. T 0 = 0, T = 7 Iitial Coditios * T = T + $ T is a sequece (f. o itegers). Solve for T? * is a recurrece,
More informationCenter, Spread, and Shape in Inference: Claims, Caveats, and Insights
Ceter, Spread, ad Shape i Iferece: Claims, Caveats, ad Isights Dr. Nacy Pfeig (Uiversity of Pittsburgh) AMATYC November 2008 Prelimiary Activities 1. I would like to produce a iterval estimate for the
More informationStatistical inference: example 1. Inferential Statistics
Statistical iferece: example 1 Iferetial Statistics POPULATION SAMPLE A clothig store chai regularly buys from a supplier large quatities of a certai piece of clothig. Each item ca be classified either
More informationCS103X: Discrete Structures Homework 4 Solutions
CS103X: Discrete Structures Homewor 4 Solutios Due February 22, 2008 Exercise 1 10 poits. Silico Valley questios: a How may possible six-figure salaries i whole dollar amouts are there that cotai at least
More informationHeavy Traffic Analysis of a Simple Closed Loop Supply Chain
Heavy Traffic Aalysis of a Simple Closed Loop Supply Chai Arka Ghosh, Sarah M. Rya, Lizhi Wag, ad Aada Weerasighe April 8, 2 Abstract We cosider a closed loop supply chai where ew products are produced
More information