Lecture Chapter 6: Convergence of Random Sequences
|
|
- Crystal Joseph
- 5 years ago
- Views:
Transcription
1 ECE5: Aalysis of Radom Sigals Fall 6 Lecture Chapter 6: Covergece of Radom Sequeces Dr Salim El Rouayheb Scribe: Abhay Ashutosh Doel, Qibo Zhag, Peiwe Tia, Pegzhe Wag, Lu Liu Radom sequece Defiitio A ifiite sequece X, =,,, of radom variables is called a radom sequece Covergece of a radom sequece Example Cosider the sequece of real umbers X =, =,,, + This sequece coverges to the limit l = We write lim X = l = This meas that i ay eighbourhood aroud we ca trap the sequece, ie, ɛ >, (ɛ) st for (ɛ) X l ɛ We ca pick ɛ to be very small ad make sure that the sequece will be trapped after reachig (ɛ) Therefore as ɛ decreases (ɛ) will icrease For example, i the cosidered sequece: Almost sure covergece ɛ =, (ɛ) =, ɛ =, (ɛ) = Defiitio A radom sequece X, =,,, 3,, coverges almost surely, or with probability oe, to the radom variable X iff P ( lim X = X) = We write X X
2 Example Let be a radom variable that is uiformly distributed o, ] Defie the radom sequece X as X = So X =, X =, X =, X 3 = 3, Let us take specific values of For istace, if = X =, X =, X = 4, X 3 = 8, We ca thik of it as a ur cotaiig sequeces, ad at each time we draw a value of, we get a sequece of fixed umbers I the example of tossig a coi, the output will be either heads or tails Whereas, i this case the output of the experimet is a radom sequece, ie, each outcome is a sequece of ifiite umbers Questio: Does this sequece of radom variables coverge? Aswer: This sequece coverges to if with probability = P ( ) X = if = with probability = P ( = ) Sice the pdf is cotiuous, the probability P ( = a) = for ay costat a Notice that the covergece of the sequece to is possible but happes with probability Therefore, we say that X coverges almost surely to, ie, X Covergece i probability Defiitio 3 A radom sequece X coverges to the radom variable X i probability if ɛ > lim P r X X ɛ} = We write : X p X Example 3 Cosider a radom variable uiformly distributed o, ] ad the sequece X defied by: with probability X = with probability Distributed as show i Figure Notice that oly X or X 3 ca be equal to for the same value of Similarly, oly oe of X 4, X 5, X 6 ad X 7 ca be equal to for the same value of ad so o ad so forth Questio: Does this sequece coverge?
3 X7() X6() X5() X4() X3() X() X() Figure : Plot of the distributio of X () Aswer: Ituitively, the sequece will coverge to Let us take some examples to see how the sequece behave for = : for = 3 : = = = = =3 =3 =4 =4 From a calculus poit of view, these sequeces ever coverge to zero because there is always a jump showig up o matter how may zeros are precedig (Fig ); for ay : X () does ot coverge i the calculus sese Which meas also that X does ot coverge to zero almost surely () 3
4 5 X Figure : Plot of the sequece for = This sequece coverges i probability sice lim P ( X ) = ɛ > Remark The observed sequece may ot coverge i calculus sese because of the itermittet jumps ; however the frequecy of those jumps goes to zero whe goes to ifiity 3 Covergece i mea square Defiitio 4 A radom sequece X coverges to a radom variable X i mea square sese if lim E X X ] = We write: X ms X Remark I mea square covergece, ot oly the frequecy of the jumps goes to zero whe goes to ifiity; but also the eergy i the jump should go to zero Example 4 Cosider a radom variable uiformly distributed over, ], ad the sequece X () defied as: a for X () = otherwise Note that P (X = a ) = ad P (X = ) = Questio: Does this sequece coverge? 4
5 Figure 3: Plot of the sequece X () Aswer: Let us check the differet covergece criteria we have see so far Almost sure covergece: X because Covergece i probability: X because lim P (X = ) = lim P X ɛ} = (Flash Forward: almost sure covergece covergece i probability) X X X X 3 Mea Square Covergece: E X ] = a (P (X = a ) + P (X = )), = a If a = lim E X ] ms = X, If a = lim E X ] = X does ot coverge i ms to I this example, the covergece of X i the mea square sese depeds o the value of a 4 Covergece i distributio Defiitio 5 (First attempt) A radom sequece X coverges to X i distributio if whe goes to ifiity, the values of the sequece are distributed accordig to a kow distributio We say d X X Example 5 Cosider the sequece X defied as: X i B( X = ) for i = (X i + ) mod = X for i > 5
6 Questio: I which sese, if ay, does this sequece coverge? Aswer: This sequece has two outcomes depedig o the value of X : X =, X : X =, X : Almost sure covergece: X does ot coverge almost surely because the probability of every jump is always equal to Covergece i probability: X does ot coverge i probability because the frequecy of the jumps is costat equal to 3 Covergece i mea square: X does ot coverge to i mea square sese because lim X E ] = E X X + ], 4 = EX ] EX ] + 4, = + + 4, 4 Covergece i distributio: At ifiity, sice we do ot kow the value of X, each value of X ca be either or with probability Hece, ay umber X is a radom variable B( ) We say, X coverges i distributio to Beroulli( ) ad we deote it by: = X d Ber( ) Example 6 (Cetral Limit Theorem)Cosider the zero-mea, uit-variace, idepedet radom variables X, X,, X ad defie the sequece S as follows: S = X + X + + X The CLT states that S coverges i distributio to N(, ), ie, Theorem Note: Almost sure covergece Covergece i mea square } S d N(, ) Covergece i probability covergece i distributio There is o relatio betwee Almost Sure ad Mea Square Covergece The relatio is uidirectioal, ie, covergece i distributio does ot imply covergece i probability either almost sure covergece or mea square covergece 6
7 3 Covergece of a radom sequece Example : defied as: Questio: Let the radom variable U be uiformly distributed o, ] Cosider the sequece X() = ( ) U Does this sequece coverge? if yes, i what sese(s)? Aswer: Almost sure covergece: Suppose The sequece becomes U = a I fact, for ay a, ] therefore, X X = a, X = a, X 3 = a 3, X 4 = a 4, lim X =, Remark 3 X because, by defiitio, a radom sequece coverges almost surely to the radom variable X if the sequece of fuctios X coverges for all values of U except for a set of values that has a probability zero Covergece i probability: Does X? Recall from theorem 3 of lecture 7: } d ms which meas that by provig almost-sure covergece, we get directly the covergece i probability ad i distributio However, for completeess we will formally prove that X coverges to i probability To do so, we have to prove that lim P ( X ɛ) = ɛ >, lim ɛ) = ɛ > 7
8 By defiitio, Thus, X = U ( ) ( ) U lim P X ɛ = lim P ɛ, () = lim P (U ɛ), () = (3) Where equatio 3 follows from the fact that fidig U, ] 3 Covergece i mea square sese: Does X coverge to i the mea square sese? I order to aswer this questio, we eed to prove that We kow that, Hece, X ms lim E X ] = lim E X ] = lim E X = lim E = lim = lim = lim U ], ], E U ], u du, ] u 3 3 = lim 3, = 4 Covergece i distributio: Does X coverge to i distributio? The formal defiitio of covergece i distributio is the followig: Hereafter, we wat to prove that X d, X d X lim F X (x) = F X (x) Recall that the limit rv X is the costat ad therefore has the followig CDF : Sice X = ( ) U, the distributio of the X i ca be derived as followig: 8
9 Figure 4: Plot of the CDF of Remark 4 At the CDF of X will be flip-floppig betwee (if is eve) ad (if is odd) (cf figure 5) which implies that there is a discotiuity at that poit Therefore, we say that X coverges i distributio to a CDF F X (x) except at poits where F X (x) is ot cotiuous Defiitio 6 X coverges to X i distributio, ie, X] d X iff lim F X (x) = F X (x) Remark 5 It is clear here that except at poits where F X (x) is ot cotiuous lim F X (x) = F x (x) except for x = Therefore, X coverges to X i distributio We could have deduced this directly from covergece i mea square sese or almost sure covergece Theorem a) If X X X X b) If X ms X X X c) If X X X d X d) If P X Y } = for all for a radom variable Y with E Y ] <, the ms X X X X Proof The proof is omitted Remark 6 Covergece i probability allows the sequece, at, to deviate from the mea for ay value with a small probability; whereas, covergece i mea square limits the amplitude of this deviatio whe (We ca thik of it as eergy we ca ot allow a big deviatio from the mea) 9
10 CDF of U CDF of X CDF of X CDF of X 3 4 Back to real aalysis Figure 5: Plot of the CDF of U, X, X ad X 3 Defiitio 7 A sequece (x ) is Cauchy if for every ɛ, there exists a large umber N st m, > N, x m x < ɛ lim,m x m x = Claim Every Cauchy sequece is coverget Couter example Cosider the sequece X Q defied as x =, x + = x+ x The limit of this sequece is give by: l = l + l, l = l +, l = ± / Q This implies that the sequece does ot coverge i Q
11 Couter example Cosider the sequece x = / i (, ) Obviously it does ot coverge i (, ) sice the limit l = / (, ) Defiitio 8 A space where every sequece coverges is called a complete space Theorem 3 R is a complete space Proof The proof is omitted Theorem 4 Cauchy criteria for covergece of a radom sequece ] a) X X P lim x m x = m, b) X ms X c) X X lim E x m x ] = m, lim P x m x ε] = m, = ɛ Proof The proofs are omitted Example 7 Cosider the sequece of example from last lecture, X i B( X = ) for i = (X i + ) mod = X for i > Goal: Our goal is to prove that this sequece does ot coverge i mea square usig Cauchy criteria This sequece has two outcomes depedig o the value of X : X =, X : X =, X : Therefore, E X X m ] = E X] ] + E X m E Xm X ], = + E X mx ] Cosider, without loss of geerality, that m > E X X m ] = if m is odd, E X X m ] = E X] = if m is eve Hece, lim E X X m ] if m is odd, =,m if m is eve, which implies that X does ot coverge i mea square by theorem 4-b)
12 Lemma Let X be a radom sequece with E X ] < X Theorem 5 Weak law of large umbers ms X iff lim m, E X mx ] exists ad is fiite Let X, X, X 3,, X i be iid radom variables E X i ] = µ, i Let The Usig the laguage of this chapter: S = X + X + + X P S µ ɛ] S µ Theorem 6 Strog law of large umbers Let X, X, X 3,, X i be iid radom variables E X i ] = µ, i Let The Usig the laguage of this chapter: S = X + X + + X ] P lim S µ ɛ = S µ Theorem 7 Cetral limit theorem Let X, X, X 3,, X i be iid radom variables E X i ] =, i Let The Usig the laguage of this chapter: Z = X + X + + X P Z z] = z Z d N(, ) π e z dz
Distribution of Random Samples & Limit theorems
STAT/MATH 395 A - PROBABILITY II UW Witer Quarter 2017 Néhémy Lim Distributio of Radom Samples & Limit theorems 1 Distributio of i.i.d. Samples Motivatig example. Assume that the goal of a study is to
More informationLecture 19: Convergence
Lecture 19: Covergece Asymptotic approach I statistical aalysis or iferece, a key to the success of fidig a good procedure is beig able to fid some momets ad/or distributios of various statistics. I may
More informationECE 330:541, Stochastic Signals and Systems Lecture Notes on Limit Theorems from Probability Fall 2002
ECE 330:541, Stochastic Sigals ad Systems Lecture Notes o Limit Theorems from robability Fall 00 I practice, there are two ways we ca costruct a ew sequece of radom variables from a old sequece of radom
More information7.1 Convergence of sequences of random variables
Chapter 7 Limit Theorems Throughout this sectio we will assume a probability space (, F, P), i which is defied a ifiite sequece of radom variables (X ) ad a radom variable X. The fact that for every ifiite
More informationEcon 325/327 Notes on Sample Mean, Sample Proportion, Central Limit Theorem, Chi-square Distribution, Student s t distribution 1.
Eco 325/327 Notes o Sample Mea, Sample Proportio, Cetral Limit Theorem, Chi-square Distributio, Studet s t distributio 1 Sample Mea By Hiro Kasahara We cosider a radom sample from a populatio. Defiitio
More informationLecture 01: the Central Limit Theorem. 1 Central Limit Theorem for i.i.d. random variables
CSCI-B609: A Theorist s Toolkit, Fall 06 Aug 3 Lecture 0: the Cetral Limit Theorem Lecturer: Yua Zhou Scribe: Yua Xie & Yua Zhou Cetral Limit Theorem for iid radom variables Let us say that we wat to aalyze
More informationLecture 20: Multivariate convergence and the Central Limit Theorem
Lecture 20: Multivariate covergece ad the Cetral Limit Theorem Covergece i distributio for radom vectors Let Z,Z 1,Z 2,... be radom vectors o R k. If the cdf of Z is cotiuous, the we ca defie covergece
More information7.1 Convergence of sequences of random variables
Chapter 7 Limit theorems Throughout this sectio we will assume a probability space (Ω, F, P), i which is defied a ifiite sequece of radom variables (X ) ad a radom variable X. The fact that for every ifiite
More informationFall 2013 MTH431/531 Real analysis Section Notes
Fall 013 MTH431/531 Real aalysis Sectio 8.1-8. Notes Yi Su 013.11.1 1. Defiitio of uiform covergece. We look at a sequece of fuctios f (x) ad study the coverget property. Notice we have two parameters
More informationInfinite Sequences and Series
Chapter 6 Ifiite Sequeces ad Series 6.1 Ifiite Sequeces 6.1.1 Elemetary Cocepts Simply speakig, a sequece is a ordered list of umbers writte: {a 1, a 2, a 3,...a, a +1,...} where the elemets a i represet
More informationConvergence of random variables. (telegram style notes) P.J.C. Spreij
Covergece of radom variables (telegram style otes).j.c. Spreij this versio: September 6, 2005 Itroductio As we kow, radom variables are by defiitio measurable fuctios o some uderlyig measurable space
More informationSequences and Series of Functions
Chapter 6 Sequeces ad Series of Fuctios 6.1. Covergece of a Sequece of Fuctios Poitwise Covergece. Defiitio 6.1. Let, for each N, fuctio f : A R be defied. If, for each x A, the sequece (f (x)) coverges
More informationMA131 - Analysis 1. Workbook 2 Sequences I
MA3 - Aalysis Workbook 2 Sequeces I Autum 203 Cotets 2 Sequeces I 2. Itroductio.............................. 2.2 Icreasig ad Decreasig Sequeces................ 2 2.3 Bouded Sequeces..........................
More informationLecture 2: Concentration Bounds
CSE 52: Desig ad Aalysis of Algorithms I Sprig 206 Lecture 2: Cocetratio Bouds Lecturer: Shaya Oveis Ghara March 30th Scribe: Syuzaa Sargsya Disclaimer: These otes have ot bee subjected to the usual scrutiy
More informationExpectation and Variance of a random variable
Chapter 11 Expectatio ad Variace of a radom variable The aim of this lecture is to defie ad itroduce mathematical Expectatio ad variace of a fuctio of discrete & cotiuous radom variables ad the distributio
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 19 11/17/2008 LAWS OF LARGE NUMBERS II THE STRONG LAW OF LARGE NUMBERS
MASSACHUSTTS INSTITUT OF TCHNOLOGY 6.436J/5.085J Fall 2008 Lecture 9 /7/2008 LAWS OF LARG NUMBRS II Cotets. The strog law of large umbers 2. The Cheroff boud TH STRONG LAW OF LARG NUMBRS While the weak
More information1.3 Convergence Theorems of Fourier Series. k k k k. N N k 1. With this in mind, we state (without proof) the convergence of Fourier series.
.3 Covergece Theorems of Fourier Series I this sectio, we preset the covergece of Fourier series. A ifiite sum is, by defiitio, a limit of partial sums, that is, a cos( kx) b si( kx) lim a cos( kx) b si(
More information6.3 Testing Series With Positive Terms
6.3. TESTING SERIES WITH POSITIVE TERMS 307 6.3 Testig Series With Positive Terms 6.3. Review of what is kow up to ow I theory, testig a series a i for covergece amouts to fidig the i= sequece of partial
More informationSequences I. Chapter Introduction
Chapter 2 Sequeces I 2. Itroductio A sequece is a list of umbers i a defiite order so that we kow which umber is i the first place, which umber is i the secod place ad, for ay atural umber, we kow which
More information1 Lecture 2: Sequence, Series and power series (8/14/2012)
Summer Jump-Start Program for Aalysis, 202 Sog-Yig Li Lecture 2: Sequece, Series ad power series (8/4/202). More o sequeces Example.. Let {x } ad {y } be two bouded sequeces. Show lim sup (x + y ) lim
More information10.1 Sequences. n term. We will deal a. a n or a n n. ( 1) n ( 1) n 1 2 ( 1) a =, 0 0,,,,, ln n. n an 2. n term.
0. Sequeces A sequece is a list of umbers writte i a defiite order: a, a,, a, a is called the first term, a is the secod term, ad i geeral eclusively with ifiite sequeces ad so each term Notatio: the sequece
More informationIntroduction to Probability. Ariel Yadin
Itroductio to robability Ariel Yadi Lecture 2 *** Ja. 7 ***. Covergece of Radom Variables As i the case of sequeces of umbers, we would like to talk about covergece of radom variables. There are may ways
More information(A sequence also can be thought of as the list of function values attained for a function f :ℵ X, where f (n) = x n for n 1.) x 1 x N +k x N +4 x 3
MATH 337 Sequeces Dr. Neal, WKU Let X be a metric space with distace fuctio d. We shall defie the geeral cocept of sequece ad limit i a metric space, the apply the results i particular to some special
More informationSequences A sequence of numbers is a function whose domain is the positive integers. We can see that the sequence
Sequeces A sequece of umbers is a fuctio whose domai is the positive itegers. We ca see that the sequece 1, 1, 2, 2, 3, 3,... is a fuctio from the positive itegers whe we write the first sequece elemet
More information62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 +
62. Power series Defiitio 16. (Power series) Give a sequece {c }, the series c x = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + is called a power series i the variable x. The umbers c are called the coefficiets of
More informationRead carefully the instructions on the answer book and make sure that the particulars required are entered on each answer book.
THE UNIVERSITY OF WARWICK FIRST YEAR EXAMINATION: Jauary 2009 Aalysis I Time Allowed:.5 hours Read carefully the istructios o the aswer book ad make sure that the particulars required are etered o each
More informationChapter 6 Infinite Series
Chapter 6 Ifiite Series I the previous chapter we cosidered itegrals which were improper i the sese that the iterval of itegratio was ubouded. I this chapter we are goig to discuss a topic which is somewhat
More informationContinuous Functions
Cotiuous Fuctios Q What does it mea for a fuctio to be cotiuous at a poit? Aswer- I mathematics, we have a defiitio that cosists of three cocepts that are liked i a special way Cosider the followig defiitio
More informationLecture 2: April 3, 2013
TTIC/CMSC 350 Mathematical Toolkit Sprig 203 Madhur Tulsiai Lecture 2: April 3, 203 Scribe: Shubhedu Trivedi Coi tosses cotiued We retur to the coi tossig example from the last lecture agai: Example. Give,
More informationn outcome is (+1,+1, 1,..., 1). Let the r.v. X denote our position (relative to our starting point 0) after n moves. Thus X = X 1 + X 2 + +X n,
CS 70 Discrete Mathematics for CS Sprig 2008 David Wager Note 9 Variace Questio: At each time step, I flip a fair coi. If it comes up Heads, I walk oe step to the right; if it comes up Tails, I walk oe
More informationA sequence of numbers is a function whose domain is the positive integers. We can see that the sequence
Sequeces A sequece of umbers is a fuctio whose domai is the positive itegers. We ca see that the sequece,, 2, 2, 3, 3,... is a fuctio from the positive itegers whe we write the first sequece elemet as
More informationNotes 12 Asymptotic Series
ECE 6382 Fall 207 David R. Jackso otes 2 Asymptotic Series Asymptotic Series A asymptotic series (as ) is of the form a ( ) f as = 0 or f a + a a + + ( ) 2 0 2 ote the asymptotically equal to sig. The
More informationTopics. Homework Problems. MATH 301 Introduction to Analysis Chapter Four Sequences. 1. Definition of convergence of sequences.
MATH 301 Itroductio to Aalysis Chapter Four Sequeces Topics 1. Defiitio of covergece of sequeces. 2. Fidig ad provig the limit of sequeces. 3. Bouded covergece theorem: Theorem 4.1.8. 4. Theorems 4.1.13
More informationLecture 8: Convergence of transformations and law of large numbers
Lecture 8: Covergece of trasformatios ad law of large umbers Trasformatio ad covergece Trasformatio is a importat tool i statistics. If X coverges to X i some sese, we ofte eed to check whether g(x ) coverges
More informationProbability 2 - Notes 10. Lemma. If X is a random variable and g(x) 0 for all x in the support of f X, then P(g(X) 1) E[g(X)].
Probability 2 - Notes 0 Some Useful Iequalities. Lemma. If X is a radom variable ad g(x 0 for all x i the support of f X, the P(g(X E[g(X]. Proof. (cotiuous case P(g(X Corollaries x:g(x f X (xdx x:g(x
More informationChapter 6 Principles of Data Reduction
Chapter 6 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 0 Chapter 6 Priciples of Data Reductio Sectio 6. Itroductio Goal: To summarize or reduce the data X, X,, X to get iformatio about a
More informationRandom Models. Tusheng Zhang. February 14, 2013
Radom Models Tusheg Zhag February 14, 013 1 Radom Walks Let me describe the model. Radom walks are used to describe the motio of a movig particle (object). Suppose that a particle (object) moves alog the
More informationAMS570 Lecture Notes #2
AMS570 Lecture Notes # Review of Probability (cotiued) Probability distributios. () Biomial distributio Biomial Experimet: ) It cosists of trials ) Each trial results i of possible outcomes, S or F 3)
More informationLesson 10: Limits and Continuity
www.scimsacademy.com Lesso 10: Limits ad Cotiuity SCIMS Academy 1 Limit of a fuctio The cocept of limit of a fuctio is cetral to all other cocepts i calculus (like cotiuity, derivative, defiite itegrals
More informationLecture 6 Simple alternatives and the Neyman-Pearson lemma
STATS 00: Itroductio to Statistical Iferece Autum 06 Lecture 6 Simple alteratives ad the Neyma-Pearso lemma Last lecture, we discussed a umber of ways to costruct test statistics for testig a simple ull
More informationPlease do NOT write in this box. Multiple Choice. Total
Istructor: Math 0560, Worksheet Alteratig Series Jauary, 3000 For realistic exam practice solve these problems without lookig at your book ad without usig a calculator. Multiple choice questios should
More informationMATH301 Real Analysis (2008 Fall) Tutorial Note #7. k=1 f k (x) converges pointwise to S(x) on E if and
MATH01 Real Aalysis (2008 Fall) Tutorial Note #7 Sequece ad Series of fuctio 1: Poitwise Covergece ad Uiform Covergece Part I: Poitwise Covergece Defiitio of poitwise covergece: A sequece of fuctios f
More informationProbability and Random Processes
Probability ad Radom Processes Lecture 5 Probability ad radom variables The law of large umbers Mikael Skoglud, Probability ad radom processes 1/21 Why Measure Theoretic Probability? Stroger limit theorems
More informationHOMEWORK I: PREREQUISITES FROM MATH 727
HOMEWORK I: PREREQUISITES FROM MATH 727 Questio. Let X, X 2,... be idepedet expoetial radom variables with mea µ. (a) Show that for Z +, we have EX µ!. (b) Show that almost surely, X + + X (c) Fid the
More informationDiscrete Mathematics and Probability Theory Spring 2016 Rao and Walrand Note 19
CS 70 Discrete Mathematics ad Probability Theory Sprig 2016 Rao ad Walrad Note 19 Some Importat Distributios Recall our basic probabilistic experimet of tossig a biased coi times. This is a very simple
More informationSection 11.8: Power Series
Sectio 11.8: Power Series 1. Power Series I this sectio, we cosider geeralizig the cocept of a series. Recall that a series is a ifiite sum of umbers a. We ca talk about whether or ot it coverges ad i
More informationParameter, Statistic and Random Samples
Parameter, Statistic ad Radom Samples A parameter is a umber that describes the populatio. It is a fixed umber, but i practice we do ot kow its value. A statistic is a fuctio of the sample data, i.e.,
More informationAre the following series absolutely convergent? n=1. n 3. n=1 n. ( 1) n. n=1 n=1
Absolute covergece Defiitio A series P a is called absolutely coverget if the series of absolute values P a is coverget. If the terms of the series a are positive, absolute covergece is the same as covergece.
More informationCentral Limit Theorem using Characteristic functions
Cetral Limit Theorem usig Characteristic fuctios RogXi Guo MAT 477 Jauary 20, 2014 RogXi Guo (2014 Cetral Limit Theorem usig Characteristic fuctios Jauary 20, 2014 1 / 15 Itroductio study a radom variable
More informationChapter 3. Strong convergence. 3.1 Definition of almost sure convergence
Chapter 3 Strog covergece As poited out i the Chapter 2, there are multiple ways to defie the otio of covergece of a sequece of radom variables. That chapter defied covergece i probability, covergece i
More informationf X (12) = Pr(X = 12) = Pr({(6, 6)}) = 1/36
Probability Distributios A Example With Dice If X is a radom variable o sample space S, the the probablity that X takes o the value c is Similarly, Pr(X = c) = Pr({s S X(s) = c} Pr(X c) = Pr({s S X(s)
More information1 Convergence in Probability and the Weak Law of Large Numbers
36-752 Advaced Probability Overview Sprig 2018 8. Covergece Cocepts: i Probability, i L p ad Almost Surely Istructor: Alessadro Rialdo Associated readig: Sec 2.4, 2.5, ad 4.11 of Ash ad Doléas-Dade; Sec
More informationPRACTICE PROBLEMS FOR THE FINAL
PRACTICE PROBLEMS FOR THE FINAL Math 36Q Fall 25 Professor Hoh Below is a list of practice questios for the Fial Exam. I would suggest also goig over the practice problems ad exams for Exam ad Exam 2 to
More informationIt is often useful to approximate complicated functions using simpler ones. We consider the task of approximating a function by a polynomial.
Taylor Polyomials ad Taylor Series It is ofte useful to approximate complicated fuctios usig simpler oes We cosider the task of approximatig a fuctio by a polyomial If f is at least -times differetiable
More informationElements of Statistical Methods Lots of Data or Large Samples (Ch 8)
Elemets of Statistical Methods Lots of Data or Large Samples (Ch 8) Fritz Scholz Sprig Quarter 2010 February 26, 2010 x ad X We itroduced the sample mea x as the average of the observed sample values x
More informationDiscrete Mathematics for CS Spring 2007 Luca Trevisan Lecture 22
CS 70 Discrete Mathematics for CS Sprig 2007 Luca Trevisa Lecture 22 Aother Importat Distributio The Geometric Distributio Questio: A biased coi with Heads probability p is tossed repeatedly util the first
More informationTopic 9: Sampling Distributions of Estimators
Topic 9: Samplig Distributios of Estimators Course 003, 2018 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be
More informationTopic 9: Sampling Distributions of Estimators
Topic 9: Samplig Distributios of Estimators Course 003, 2018 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be
More informationThis section is optional.
4 Momet Geeratig Fuctios* This sectio is optioal. The momet geeratig fuctio g : R R of a radom variable X is defied as g(t) = E[e tx ]. Propositio 1. We have g () (0) = E[X ] for = 1, 2,... Proof. Therefore
More informationMAT1026 Calculus II Basic Convergence Tests for Series
MAT026 Calculus II Basic Covergece Tests for Series Egi MERMUT 202.03.08 Dokuz Eylül Uiversity Faculty of Sciece Departmet of Mathematics İzmir/TURKEY Cotets Mootoe Covergece Theorem 2 2 Series of Real
More informationTopic 9: Sampling Distributions of Estimators
Topic 9: Samplig Distributios of Estimators Course 003, 2016 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be
More informationRandom Variables, Sampling and Estimation
Chapter 1 Radom Variables, Samplig ad Estimatio 1.1 Itroductio This chapter will cover the most importat basic statistical theory you eed i order to uderstad the ecoometric material that will be comig
More informationMATH 312 Midterm I(Spring 2015)
MATH 3 Midterm I(Sprig 05) Istructor: Xiaowei Wag Feb 3rd, :30pm-3:50pm, 05 Problem (0 poits). Test for covergece:.. 3.. p, p 0. (coverges for p < ad diverges for p by ratio test.). ( coverges, sice (log
More informationLECTURE 8: ASYMPTOTICS I
LECTURE 8: ASYMPTOTICS I We are iterested i the properties of estimators as. Cosider a sequece of radom variables {, X 1}. N. M. Kiefer, Corell Uiversity, Ecoomics 60 1 Defiitio: (Weak covergece) A sequece
More information32 estimating the cumulative distribution function
32 estimatig the cumulative distributio fuctio 4.6 types of cofidece itervals/bads Let F be a class of distributio fuctios F ad let θ be some quatity of iterest, such as the mea of F or the whole fuctio
More informationGlivenko-Cantelli Classes
CS28B/Stat24B (Sprig 2008 Statistical Learig Theory Lecture: 4 Gliveko-Catelli Classes Lecturer: Peter Bartlett Scribe: Michelle Besi Itroductio This lecture will cover Gliveko-Catelli (GC classes ad itroduce
More information4. Partial Sums and the Central Limit Theorem
1 of 10 7/16/2009 6:05 AM Virtual Laboratories > 6. Radom Samples > 1 2 3 4 5 6 7 4. Partial Sums ad the Cetral Limit Theorem The cetral limit theorem ad the law of large umbers are the two fudametal theorems
More informationLecture 12: November 13, 2018
Mathematical Toolkit Autum 2018 Lecturer: Madhur Tulsiai Lecture 12: November 13, 2018 1 Radomized polyomial idetity testig We will use our kowledge of coditioal probability to prove the followig lemma,
More informationLecture 4. We also define the set of possible values for the random walk as the set of all x R d such that P(S n = x) > 0 for some n.
Radom Walks ad Browia Motio Tel Aviv Uiversity Sprig 20 Lecture date: Mar 2, 20 Lecture 4 Istructor: Ro Peled Scribe: Lira Rotem This lecture deals primarily with recurrece for geeral radom walks. We preset
More informationMath Solutions to homework 6
Math 175 - Solutios to homework 6 Cédric De Groote November 16, 2017 Problem 1 (8.11 i the book): Let K be a compact Hermitia operator o a Hilbert space H ad let the kerel of K be {0}. Show that there
More informationDiscrete Mathematics for CS Spring 2008 David Wagner Note 22
CS 70 Discrete Mathematics for CS Sprig 2008 David Wager Note 22 I.I.D. Radom Variables Estimatig the bias of a coi Questio: We wat to estimate the proportio p of Democrats i the US populatio, by takig
More information6 Infinite random sequences
Tel Aviv Uiversity, 2006 Probability theory 55 6 Ifiite radom sequeces 6a Itroductory remarks; almost certaity There are two mai reasos for eterig cotiuous probability: ifiitely high resolutio; edless
More informationSequences. A Sequence is a list of numbers written in order.
Sequeces A Sequece is a list of umbers writte i order. {a, a 2, a 3,... } The sequece may be ifiite. The th term of the sequece is the th umber o the list. O the list above a = st term, a 2 = 2 d term,
More information2.1. Convergence in distribution and characteristic functions.
3 Chapter 2. Cetral Limit Theorem. Cetral limit theorem, or DeMoivre-Laplace Theorem, which also implies the wea law of large umbers, is the most importat theorem i probability theory ad statistics. For
More informationLet us give one more example of MLE. Example 3. The uniform distribution U[0, θ] on the interval [0, θ] has p.d.f.
Lecture 5 Let us give oe more example of MLE. Example 3. The uiform distributio U[0, ] o the iterval [0, ] has p.d.f. { 1 f(x =, 0 x, 0, otherwise The likelihood fuctio ϕ( = f(x i = 1 I(X 1,..., X [0,
More informationM17 MAT25-21 HOMEWORK 5 SOLUTIONS
M17 MAT5-1 HOMEWORK 5 SOLUTIONS 1. To Had I Cauchy Codesatio Test. Exercise 1: Applicatio of the Cauchy Codesatio Test Use the Cauchy Codesatio Test to prove that 1 diverges. Solutio 1. Give the series
More informationECE 6980 An Algorithmic and Information-Theoretic Toolbox for Massive Data
ECE 6980 A Algorithmic ad Iformatio-Theoretic Toolbo for Massive Data Istructor: Jayadev Acharya Lecture # Scribe: Huayu Zhag 8th August, 017 1 Recap X =, ε is a accuracy parameter, ad δ is a error parameter.
More informationNotes for Lecture 11
U.C. Berkeley CS78: Computatioal Complexity Hadout N Professor Luca Trevisa 3/4/008 Notes for Lecture Eigevalues, Expasio, ad Radom Walks As usual by ow, let G = (V, E) be a udirected d-regular graph with
More informationEE 4TM4: Digital Communications II Probability Theory
1 EE 4TM4: Digital Commuicatios II Probability Theory I. RANDOM VARIABLES A radom variable is a real-valued fuctio defied o the sample space. Example: Suppose that our experimet cosists of tossig two fair
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 6 9/23/2013. Brownian motion. Introduction
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/5.070J Fall 203 Lecture 6 9/23/203 Browia motio. Itroductio Cotet.. A heuristic costructio of a Browia motio from a radom walk. 2. Defiitio ad basic properties
More informationMATH4822E FOURIER ANALYSIS AND ITS APPLICATIONS
MATH48E FOURIER ANALYSIS AND ITS APPLICATIONS 7.. Cesàro summability. 7. Summability methods Arithmetic meas. The followig idea is due to the Italia geometer Eresto Cesàro (859-96). He shows that eve if
More informationJanuary 25, 2017 INTRODUCTION TO MATHEMATICAL STATISTICS
Jauary 25, 207 INTRODUCTION TO MATHEMATICAL STATISTICS Abstract. A basic itroductio to statistics assumig kowledge of probability theory.. Probability I a typical udergraduate problem i probability, we
More informationMath 341 Lecture #31 6.5: Power Series
Math 341 Lecture #31 6.5: Power Series We ow tur our attetio to a particular kid of series of fuctios, amely, power series, f(x = a x = a 0 + a 1 x + a 2 x 2 + where a R for all N. I terms of a series
More informationSUMMARY OF SEQUENCES AND SERIES
SUMMARY OF SEQUENCES AND SERIES Importat Defiitios, Results ad Theorems for Sequeces ad Series Defiitio. A sequece {a } has a limit L ad we write lim a = L if for every ɛ > 0, there is a correspodig iteger
More informationMa 530 Infinite Series I
Ma 50 Ifiite Series I Please ote that i additio to the material below this lecture icorporated material from the Visual Calculus web site. The material o sequeces is at Visual Sequeces. (To use this li
More informationMATH 112: HOMEWORK 6 SOLUTIONS. Problem 1: Rudin, Chapter 3, Problem s k < s k < 2 + s k+1
MATH 2: HOMEWORK 6 SOLUTIONS CA PRO JIRADILOK Problem. If s = 2, ad Problem : Rudi, Chapter 3, Problem 3. s + = 2 + s ( =, 2, 3,... ), prove that {s } coverges, ad that s < 2 for =, 2, 3,.... Proof. The
More informationUNIT 2 DIFFERENT APPROACHES TO PROBABILITY THEORY
UNIT 2 DIFFERENT APPROACHES TO PROBABILITY THEORY Structure 2.1 Itroductio Objectives 2.2 Relative Frequecy Approach ad Statistical Probability 2. Problems Based o Relative Frequecy 2.4 Subjective Approach
More informationDiscrete Mathematics and Probability Theory Summer 2014 James Cook Note 15
CS 70 Discrete Mathematics ad Probability Theory Summer 2014 James Cook Note 15 Some Importat Distributios I this ote we will itroduce three importat probability distributios that are widely used to model
More informationMachine Learning Theory (CS 6783)
Machie Learig Theory (CS 6783) Lecture 2 : Learig Frameworks, Examples Settig up learig problems. X : istace space or iput space Examples: Computer Visio: Raw M N image vectorized X = 0, 255 M N, SIFT
More informationAMS 216 Stochastic Differential Equations Lecture 02 Copyright by Hongyun Wang, UCSC ( ( )) 2 = E X 2 ( ( )) 2
AMS 216 Stochastic Differetial Equatios Lecture 02 Copyright by Hogyu Wag, UCSC Review of probability theory (Cotiued) Variace: var X We obtai: = E X E( X ) 2 = E( X 2 ) 2E ( X )E X var( X ) = E X 2 Stadard
More informationQuiz No. 1. ln n n. 1. Define: an infinite sequence A function whose domain is N 2. Define: a convergent sequence A sequence that has a limit
Quiz No.. Defie: a ifiite sequece A fuctio whose domai is N 2. Defie: a coverget sequece A sequece that has a limit 3. Is this sequece coverget? Why or why ot? l Yes, it is coverget sice L=0 by LHR. INFINITE
More informationOn a Smarandache problem concerning the prime gaps
O a Smaradache problem cocerig the prime gaps Felice Russo Via A. Ifate 7 6705 Avezzao (Aq) Italy felice.russo@katamail.com Abstract I this paper, a problem posed i [] by Smaradache cocerig the prime gaps
More informationSTAT Homework 1 - Solutions
STAT-36700 Homework 1 - Solutios Fall 018 September 11, 018 This cotais solutios for Homework 1. Please ote that we have icluded several additioal commets ad approaches to the problems to give you better
More information1 The Haar functions and the Brownian motion
1 The Haar fuctios ad the Browia motio 1.1 The Haar fuctios ad their completeess The Haar fuctios The basic Haar fuctio is 1 if x < 1/2, ψx) = 1 if 1/2 x < 1, otherwise. 1.1) It has mea zero 1 ψx)dx =,
More informationMath 10A final exam, December 16, 2016
Please put away all books, calculators, cell phoes ad other devices. You may cosult a sigle two-sided sheet of otes. Please write carefully ad clearly, USING WORDS (ot just symbols). Remember that the
More informationMA131 - Analysis 1. Workbook 3 Sequences II
MA3 - Aalysis Workbook 3 Sequeces II Autum 2004 Cotets 2.8 Coverget Sequeces........................ 2.9 Algebra of Limits......................... 2 2.0 Further Useful Results........................
More informationChapter 2 The Monte Carlo Method
Chapter 2 The Mote Carlo Method The Mote Carlo Method stads for a broad class of computatioal algorithms that rely o radom sampligs. It is ofte used i physical ad mathematical problems ad is most useful
More informationMA Advanced Econometrics: Properties of Least Squares Estimators
MA Advaced Ecoometrics: Properties of Least Squares Estimators Karl Whela School of Ecoomics, UCD February 5, 20 Karl Whela UCD Least Squares Estimators February 5, 20 / 5 Part I Least Squares: Some Fiite-Sample
More informationLarge Sample Theory. Convergence. Central Limit Theorems Asymptotic Distribution Delta Method. Convergence in Probability Convergence in Distribution
Large Sample Theory Covergece Covergece i Probability Covergece i Distributio Cetral Limit Theorems Asymptotic Distributio Delta Method Covergece i Probability A sequece of radom scalars {z } = (z 1,z,
More informationf X (12) = Pr(X = 12) = Pr({(6, 6)}) = 1/36
Probability Distributios A Example With Dice If X is a radom variable o sample space S, the the probability that X takes o the value c is Similarly, Pr(X = c) = Pr({s S X(s) = c}) Pr(X c) = Pr({s S X(s)
More information