An alternative proof of a theorem of Aldous. concerning convergence in distribution for martingales.
|
|
- Benjamin Emory Perry
- 6 years ago
- Views:
Transcription
1 A alterative proof of a theorem of Aldous cocerig covergece i distributio for martigales. Maurizio Pratelli Dipartimeto di Matematica, Uiversità di Pisa. Via Buoarroti 2. I Pisa, Italy pratelli@dm.uipi.it We cosider regular right cotiuous stochastic processes X = (X t 0 t 1 defied o the fiite time iterval [0, 1: let P X be the distributio of X o the caoical Skorokhod space D = D([0, 1; R of càdlàg paths. We cosider o D, besides the usual Skorokhod topology referred as S topology (Jacod Shiryaev is perhaps the best referece for our purposes, see [4, the pseudopath or MZ topology: we refer to the paper of Meyer Zheg ([6 for a complete accout of this rather eglected topology (see also Kurtz [5. We will use the otatio X = S X (respectively X = MZ X to idicate that the probabilities P X coverge strictly to P X whe the space D is edowed respectively with the S or the MZ topology. We will write also X = f.d.d. X to idicate that all fiite dimesioal distributios of (Xt 0 t 1 coverge to those of (X t 0 t 1. The followig theorem holds true: Theorem. Let (M be a sequece of martigales, ad M a cotiuous martigale, ad suppose that the followig itegrability coditio is satisfied: (1 all radom variables ( sup 0 t 1 M t, = 1, 2,... are uiformly itegrable. The the followig statemets are equivalet: (a M = S M, (b M = f.d.d. M, (c M = MZ M. The implicatio (a= (b is quite obvious, sice Skorokhod covergece implies covergece of fiite dimesioal distributios for all cotiuity poits of M (see [4.
2 The implicatios (b= (c is a easy cosequece of the results of Meyer Zheg: i fact the sequece (M is tight for the MZ topology ([6 p. 368 ad, if X is a limit process, there exists ([6 p. 365 a subsequece (M k ad a set I [0, 1 of full Lebesgue measure such that all fiite dimesioal distributios (M k t t I coverge to those of (X t t I : ecessarily P X = P M. Aldous (i [2 gives a proof of the implicatio (b = (a, but (although he does ot metio the MZ topology the implicatio (c = (a is more or less implicit i his paper (see [2 p The purpose of this paper is to give a proof of the implicatio (c = (a, completely differet form the Aldous origial oe ad strictly i the spirit of the paper of Meyer Zheg; I hope that this cotributes also to a better kowledge of the result of Stoppig times ad tightess II ([2, which is i my opiio very importat ad seems to be almost ukow. The proof will be postpoed after some remarks. Remark 1. I wat to poit out that Aldous proof of the implicatio (b = (a requires the followig weaker itegrability coditio: (2 all radom variables M1, = 1, 2,... are uiformly itegrable. Coditio (2 implies that all r.v. of the form MT, = 1, 2,..., with T a atural stoppig time for M, are uiformly itegrable; istead our proof eeds a more striget coditio, i.e. that all r.v. of the form MS, = 1, 2,..., with S a radom variable i [0, 1, are uiformly itegrable. Remark 2. The extesio of the Theorem to processes whose time iterval is [0, + is straightforward: i that case the correct hypothesis is that, for every fixed t, the r.v. sup 0 s t Ms, = 1, 2,... are uiformly itegrable. I fact, if the limit fuctio f is cotiuous, f f for the S topology (respectively the MZ topology o D(R + ; R if ad oly if the restrictios of f to every fiite time iterval coverge to those of f (for the S or the MZ topology. Remark 3. The Theorem fails to be true if the limit martigale M is ot cotiuous ([2 p. 588, ad fails for more geeral processes, e.g. for supermartigales. Let ideed T be a Poisso r.v. ad put, for every : X t = ( I {t T } t T ( (t T I {t T } 1. The processes X are supermartigales whose paths coverge i measure (but ot uiformly to the paths of the cotiuous supermartigale X t = (t T. Remark 4. Suppose that the processes X are supermartigales, ad cosider their Doob Meyer decompositios X = M A. If separately M = MZ M ad the martigale M is cotiuous, ad if A = MZ A ad the icreasig process A is cotiuous, the X = S X = M A (remark that, for mootoe processes, covergece i the MZ sese to a cotiuous limit implies covergece for the S topology. A applicatio of the latter result ca be foud i [7, theorem 5.5.
3 The proof of the implicatio (c = (a of the Theorem is rather techical, ad will be divided i several steps. Step 1. Give ɛ > 0, there exists δ > 0 such that, if S is a r.v. with values i [0, 1 ad 0 d δ : (3 E [ M S+d M S ɛ. This is a easy cosequece of the path-cotiuity of the limit process M, ad of the itegrability of M = sup 0 t 1 M t. Remark that the fuctio f sup 0 t 1 f(t is lower semi-cotiuous o D edowed with the topology of covergece i measure (i.e. the MZ topology; therefore the itegrability of M is a cosequece of coditio (1 of the theorem. Step 2. Suppose that (a is false; the the sequece does ot verify Aldous tightess coditio ([1 p. 335, see also [4; therefore there exists ɛ > 0 such that for every δ > 0 it is possible to determie a subsequece k ad, for every k, a atural stoppig time T k (i.e. a stoppig time for the filtratio geerated by M k ad 0 < d k δ such that (4 E [ k M k T k +d k M k T k ɛ. (I the sequel, for the sake of simplicity of otatios, we will assume that idices have bee reamed so that the whole sequece verifies (4. We choose δ such that, for ay r.v. S whatsoever, we also have (step 1 E [ M S+2δ M S ɛ 4. Step 3. There exists a radom variable T with values i [0,1 such that (M, T coverge i distributio to (M, T o the space D([0, 1, R + [0, 1 equipped with the product topology (D beig equipped with the MZ topology. I fact the laws of (M, T are evidetly tight sice the laws of M are tight o D ([6 p. 368; we poit out that the limit r.v. T is ot a atural stoppig time for the stochastic process M (but it ca be proved that M is a martigale for the caoical filtratio o D [0, 1, i.e. the smallest filtratio that makes M adapted ad T a stoppig time. Step 4. For c ad d i [0, 1, we have the iequality (5 E [ M T +δ+c MT d ε 2. (It is techically coveiet to regard each process M as exteded to [ 1, 2 by puttig M t = M 0 for t < 0 ad M t = M 1 for t > 1 : this eables us to write M T +δ istead of M (T +δ 1. Cocerig the iequality (5, firstly we ote that ( M T +d M T = E [ M T +δ+c M T FT +d ad therefore E [ M T +δ+c M T E [ M T +d M T ɛ.
4 The we remark that (T c is ot a stoppig time, but the r.v. MT c is F T measurable: i fact MT c.i{t t} = M(T t c.i {T t} ad (T t c is F t measurable. Let X = ( MT +δ+c M ( T, Y = M T MT c ad G = FT : Y is G- adapted ad E[X G = 0. We remark that E [X + G = E [X G = 1 2 E [ X G, ad that X + Y X +.I {Y 0} + X.I {Y <0}. Oe gets E [ X + Y G 1 2 E[ X G ; ad, takig the expectatios, the iequality (5. Step 5. There exists a subsequece ad a set I [ 1, 1 of full Lebesgue measure such that the fiite dimesioal distributios of ( MT +t coverge to those of t I (M T +t t I. The proof of this step is a slight modificatio of the argumet give i [6 (p. 364: Dudley s extesio of the Skorokhod represetatio theorem implies that oe ca fid o some probability space (Ω, F, P some radom variables (X, S ad (X, S with values i D [0, 1 such that the laws of (X, S (resp. (X, S are equal to those of (M, T (resp. (M, T ad that, for almost all ω, (X (ω, S (ω coverge to (X(ω, S(ω : to be accurate, the paths t (Xt (ω coverge i measure to the path t (X t (ω ad S (ω coverge to S(ω. We remark that the Skorokhod theorem caot be applied directly sice D is ot a Polish space ([6 p. 372, but Dudley s extesio works well (see [3. By substitutig X with arctg(x, we ca suppose that X ad X are uiformly bouded: therefore we have ( +1 (6 lim dt X T (ω+t (ω X T (ω+t(ω dp(ω = 0. 1 Ω By takig a subsequece, we fid that for every t i a set I [ 1, 1 of full Lebesgue measure, (7 lim X T (ω+t (ω X T (ω+t(ω dp(ω = 0. Ω Hece oe gets easily the covergece of fiite dimesioal distributios of ( M T +t t I. Step 6. We choose 0 d, c 1 such that d + c < δ ad that ( MT +δ+c, M T d coverge i distributio to (M T +δ+c, M T d ; sice the r.v. ivolved are uiformly itegrable, the iequality (5 gives i the limit ad fially we have a cotradictio. E [ M T +δ+c M T d ε 2
5 Refereces [1 Aldous D.: Stoppig Times ad Tightess. A. Prob. 6, (1979 [2 Aldous D.: Stoppig Times ad Tightess II. A. Prob. 17, (1989 [3 Dudley R.M.: Distaces of Probability measures ad Radom Variables. A. of Math. Stat. 39, (1968 [4 Jacod J., Shiryaev A.N.: Limit theorems for stochastic processes. Berli, Heidelberg, New York: Spriger [5 Kurtz T.G.: Radom time chages ad covergece i distributio uder the Meyer Zheg coditios A. Prob. 19, (1991 [6 Meyer P.A., Zheg W.A.: Tigthess criteria for laws of semimartigales. A. Ist. Heri Poicaré Vol. 20 No. 4, (1984 [7 Muliacci S., Pratelli M.: Fuctioal covergece of Sell evelopes: Applicatios to America optios approximatios. Fiace Stochast. 2, (1998
Definition 4.2. (a) A sequence {x n } in a Banach space X is a basis for X if. unique scalars a n (x) such that x = n. a n (x) x n. (4.
4. BASES I BAACH SPACES 39 4. BASES I BAACH SPACES Sice a Baach space X is a vector space, it must possess a Hamel, or vector space, basis, i.e., a subset {x γ } γ Γ whose fiite liear spa is all of X ad
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 informationSolutions to HW Assignment 1
Solutios to HW: 1 Course: Theory of Probability II Page: 1 of 6 Uiversity of Texas at Austi Solutios to HW Assigmet 1 Problem 1.1. Let Ω, F, {F } 0, P) be a filtered probability space ad T a stoppig time.
More informationNotes on Snell Envelops and Examples
Notes o Sell Evelops ad Examples Example (Secretary Problem): Coside a pool of N cadidates whose qualificatios are represeted by ukow umbers {a > a 2 > > a N } from best to last. They are iterviewed sequetially
More informationProduct measures, Tonelli s and Fubini s theorems For use in MAT3400/4400, autumn 2014 Nadia S. Larsen. Version of 13 October 2014.
Product measures, Toelli s ad Fubii s theorems For use i MAT3400/4400, autum 2014 Nadia S. Larse Versio of 13 October 2014. 1. Costructio of the product measure The purpose of these otes is to preset the
More informationLecture 3 : Random variables and their distributions
Lecture 3 : Radom variables ad their distributios 3.1 Radom variables Let (Ω, F) ad (S, S) be two measurable spaces. A map X : Ω S is measurable or a radom variable (deoted r.v.) if X 1 (A) {ω : X(ω) A}
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 informationNotes 19 : Martingale CLT
Notes 9 : Martigale CLT Math 733-734: Theory of Probability Lecturer: Sebastie Roch Refereces: [Bil95, Chapter 35], [Roc, Chapter 3]. Sice we have ot ecoutered weak covergece i some time, we first recall
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 21 11/27/2013
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 21 11/27/2013 Fuctioal Law of Large Numbers. Costructio of the Wieer Measure Cotet. 1. Additioal techical results o weak covergece
More informationINFINITE SEQUENCES AND SERIES
INFINITE SEQUENCES AND SERIES INFINITE SEQUENCES AND SERIES I geeral, it is difficult to fid the exact sum of a series. We were able to accomplish this for geometric series ad the series /[(+)]. This is
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 informationLecture 3 The Lebesgue Integral
Lecture 3: The Lebesgue Itegral 1 of 14 Course: Theory of Probability I Term: Fall 2013 Istructor: Gorda Zitkovic Lecture 3 The Lebesgue Itegral The costructio of the itegral Uless expressly specified
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 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 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 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 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 information7 Sequences of real numbers
40 7 Sequeces of real umbers 7. Defiitios ad examples Defiitio 7... A sequece of real umbers is a real fuctio whose domai is the set N of atural umbers. Let s : N R be a sequece. The the values of s are
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 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 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 informationChapter 7 Isoperimetric problem
Chapter 7 Isoperimetric problem Recall that the isoperimetric problem (see the itroductio its coectio with ido s proble) is oe of the most classical problem of a shape optimizatio. It ca be formulated
More informationINFINITE SEQUENCES AND SERIES
11 INFINITE SEQUENCES AND SERIES INFINITE SEQUENCES AND SERIES 11.4 The Compariso Tests I this sectio, we will lear: How to fid the value of a series by comparig it with a kow series. COMPARISON TESTS
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 informationLecture Notes for Analysis Class
Lecture Notes for Aalysis Class Topological Spaces A topology for a set X is a collectio T of subsets of X such that: (a) X ad the empty set are i T (b) Uios of elemets of T are i T (c) Fiite itersectios
More informationMeasure and Measurable Functions
3 Measure ad Measurable Fuctios 3.1 Measure o a Arbitrary σ-algebra Recall from Chapter 2 that the set M of all Lebesgue measurable sets has the followig properties: R M, E M implies E c M, E M for N implies
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 informationy X F n (y), To see this, let y Y and apply property (ii) to find a sequence {y n } X such that y n y and lim sup F n (y n ) F (y).
Modica Mortola Fuctioal 2 Γ-Covergece Let X, d) be a metric space ad cosider a sequece {F } of fuctioals F : X [, ]. We say that {F } Γ-coverges to a fuctioal F : X [, ] if the followig properties hold:
More informationNotes 5 : More on the a.s. convergence of sums
Notes 5 : More o the a.s. covergece of sums Math 733-734: Theory of Probability Lecturer: Sebastie Roch Refereces: Dur0, Sectios.5; Wil9, Sectio 4.7, Shi96, Sectio IV.4, Dur0, Sectio.. Radom series. Three-series
More informationMAS111 Convergence and Continuity
MAS Covergece ad Cotiuity Key Objectives At the ed of the course, studets should kow the followig topics ad be able to apply the basic priciples ad theorems therei to solvig various problems cocerig covergece
More informationAsymptotics of Predictive Distributions
Asymptotics of redictive Distributios atrizia Berti, Luca ratelli ad ietro Rigo Abstract Let (X ) be a sequece of radom variables, adapted to a filtratio (G ), ad let μ = (1/) i=1 δ X i ad a ( ) = (X +1
More informationMath 113 Exam 4 Practice
Math Exam 4 Practice Exam 4 will cover.-.. This sheet has three sectios. The first sectio will remid you about techiques ad formulas that you should kow. The secod gives a umber of practice questios for
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 informationf n (x) f m (x) < ɛ/3 for all x A. By continuity of f n and f m we can find δ > 0 such that d(x, x 0 ) < δ implies that
Lecture 15 We have see that a sequece of cotiuous fuctios which is uiformly coverget produces a limit fuctio which is also cotiuous. We shall stregthe this result ow. Theorem 1 Let f : X R or (C) be a
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 informationANSWERS TO MIDTERM EXAM # 2
MATH 03, FALL 003 ANSWERS TO MIDTERM EXAM # PENN STATE UNIVERSITY Problem 1 (18 pts). State ad prove the Itermediate Value Theorem. Solutio See class otes or Theorem 5.6.1 from our textbook. Problem (18
More informationMath 113 Exam 3 Practice
Math Exam Practice Exam will cover.-.9. This sheet has three sectios. The first sectio will remid you about techiques ad formulas that you should kow. The secod gives a umber of practice questios for you
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 informationSOLUTIONS TO EXAM 3. Solution: Note that this defines two convergent geometric series with respective radii r 1 = 2/5 < 1 and r 2 = 1/5 < 1.
SOLUTIONS TO EXAM 3 Problem Fid the sum of the followig series 2 + ( ) 5 5 2 5 3 25 2 2 This series diverges Solutio: Note that this defies two coverget geometric series with respective radii r 2/5 < ad
More informationSelf-normalized deviation inequalities with application to t-statistic
Self-ormalized deviatio iequalities with applicatio to t-statistic Xiequa Fa Ceter for Applied Mathematics, Tiaji Uiversity, 30007 Tiaji, Chia Abstract Let ξ i i 1 be a sequece of idepedet ad symmetric
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 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 informationEntropy Rates and Asymptotic Equipartition
Chapter 29 Etropy Rates ad Asymptotic Equipartitio Sectio 29. itroduces the etropy rate the asymptotic etropy per time-step of a stochastic process ad shows that it is well-defied; ad similarly for iformatio,
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 informationEmpirical Processes: Glivenko Cantelli Theorems
Empirical Processes: Gliveko Catelli Theorems Mouliath Baerjee Jue 6, 200 Gliveko Catelli classes of fuctios The reader is referred to Chapter.6 of Weller s Torgo otes, Chapter??? of VDVW ad Chapter 8.3
More information4x 2. (n+1) x 3 n+1. = lim. 4x 2 n+1 n3 n. n 4x 2 = lim = 3
Exam Problems (x. Give the series (, fid the values of x for which this power series coverges. Also =0 state clearly what the radius of covergece is. We start by settig up the Ratio Test: x ( x x ( x x
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 informationMathematical Methods for Physics and Engineering
Mathematical Methods for Physics ad Egieerig Lecture otes Sergei V. Shabaov Departmet of Mathematics, Uiversity of Florida, Gaiesville, FL 326 USA CHAPTER The theory of covergece. Numerical sequeces..
More informationSTAT331. Example of Martingale CLT with Cox s Model
STAT33 Example of Martigale CLT with Cox s Model I this uit we illustrate the Martigale Cetral Limit Theorem by applyig it to the partial likelihood score fuctio from Cox s model. For simplicity of presetatio
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 informationTR/46 OCTOBER THE ZEROS OF PARTIAL SUMS OF A MACLAURIN EXPANSION A. TALBOT
TR/46 OCTOBER 974 THE ZEROS OF PARTIAL SUMS OF A MACLAURIN EXPANSION by A. TALBOT .. Itroductio. A problem i approximatio theory o which I have recetly worked [] required for its solutio a proof that the
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 3 9/11/2013. Large deviations Theory. Cramér s Theorem
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/5.070J Fall 203 Lecture 3 9//203 Large deviatios Theory. Cramér s Theorem Cotet.. Cramér s Theorem. 2. Rate fuctio ad properties. 3. Chage of measure techique.
More informationsin(n) + 2 cos(2n) n 3/2 3 sin(n) 2cos(2n) n 3/2 a n =
60. Ratio ad root tests 60.1. Absolutely coverget series. Defiitio 13. (Absolute covergece) A series a is called absolutely coverget if the series of absolute values a is coverget. The absolute covergece
More informationFourier Series and their Applications
Fourier Series ad their Applicatios The fuctios, cos x, si x, cos x, si x, are orthogoal over (, ). m cos mx cos xdx = m = m = = cos mx si xdx = for all m, { m si mx si xdx = m = I fact the fuctios satisfy
More informationDANIELL AND RIEMANN INTEGRABILITY
DANIELL AND RIEMANN INTEGRABILITY ILEANA BUCUR We itroduce the otio of Riema itegrable fuctio with respect to a Daiell itegral ad prove the approximatio theorem of such fuctios by a mootoe sequece of Jorda
More informationMATH 413 FINAL EXAM. f(x) f(y) M x y. x + 1 n
MATH 43 FINAL EXAM Math 43 fial exam, 3 May 28. The exam starts at 9: am ad you have 5 miutes. No textbooks or calculators may be used durig the exam. This exam is prited o both sides of the paper. Good
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 informationSeunghee Ye Ma 8: Week 5 Oct 28
Week 5 Summary I Sectio, we go over the Mea Value Theorem ad its applicatios. I Sectio 2, we will recap what we have covered so far this term. Topics Page Mea Value Theorem. Applicatios of the Mea Value
More informationSEMIGROUPS. D. Pfeifer. Communicated by Jerome A. Goldstein Dedicated to E.S. Lyapin on his 70th Birthday
A NOTE ON ~ROBABILISTIC REPRESENTATIONS OF OPERATOR SEMIGROUPS D. Pfeifer Commuicated by Jerome A. Goldstei Dedicated to E.S. Lyapi o his 70th Birthday I the theory of strogly cotiuous semigroups of bouded
More informationStochastic Integration and Ito s Formula
CHAPTER 3 Stochastic Itegratio ad Ito s Formula I this chapter we discuss Itô s theory of stochastic itegratio. This is a vast subject. However, our goal is rather modest: we will develop this theory oly
More informationAdvanced Analysis. Min Yan Department of Mathematics Hong Kong University of Science and Technology
Advaced Aalysis Mi Ya Departmet of Mathematics Hog Kog Uiversity of Sciece ad Techology September 3, 009 Cotets Limit ad Cotiuity 7 Limit of Sequece 8 Defiitio 8 Property 3 3 Ifiity ad Ifiitesimal 8 4
More informationApproximation by Superpositions of a Sigmoidal Function
Zeitschrift für Aalysis ud ihre Aweduge Joural for Aalysis ad its Applicatios Volume 22 (2003, No. 2, 463 470 Approximatio by Superpositios of a Sigmoidal Fuctio G. Lewicki ad G. Mario Abstract. We geeralize
More informationKolmogorov-Smirnov type Tests for Local Gaussianity in High-Frequency Data
Proceedigs 59th ISI World Statistics Cogress, 5-30 August 013, Hog Kog (Sessio STS046) p.09 Kolmogorov-Smirov type Tests for Local Gaussiaity i High-Frequecy Data George Tauche, Duke Uiversity Viktor Todorov,
More informationMath 140A Elementary Analysis Homework Questions 3-1
Math 0A Elemetary Aalysis Homework Questios -.9 Limits Theorems for Sequeces Suppose that lim x =, lim y = 7 ad that all y are o-zero. Detarime the followig limits: (a) lim(x + y ) (b) lim y x y Let s
More informationAn introduction to stochastic integration with respect to continuous semimartingales
A itroductio to stochastic itegratio with respect to cotiuous semimartigales Alexader Sool Departmet of Mathematical Scieces Uiversity of Copehage Departmet of Mathematical Scieces Uiversity of Copehage
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 informationErgodicity of Stochastic Processes and the Markov Chain Central Limit Theorem
Uiversity of Bristol School of Mathematics Ergodicity of Stochastic Processes ad the Markov Chai Cetral Limit Theorem A Project of 30 Credit Poits at Level 7 For the Degree of MSci Mathematics Author:
More informationIntegrable Functions. { f n } is called a determining sequence for f. If f is integrable with respect to, then f d does exist as a finite real number
MATH 532 Itegrable Fuctios Dr. Neal, WKU We ow shall defie what it meas for a measurable fuctio to be itegrable, show that all itegral properties of simple fuctios still hold, ad the give some coditios
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 informationThe Poisson Summation Formula and an Application to Number Theory Jason Payne Math 248- Introduction Harmonic Analysis, February 18, 2010
The Poisso Summatio Formula ad a Applicatio to Number Theory Jaso Paye Math 48- Itroductio Harmoic Aalysis, February 8, This talk will closely follow []; however some material has bee adapted to a slightly
More informationFUNDAMENTALS OF REAL ANALYSIS by
FUNDAMENTALS OF REAL ANALYSIS by Doğa Çömez Backgroud: All of Math 450/1 material. Namely: basic set theory, relatios ad PMI, structure of N, Z, Q ad R, basic properties of (cotiuous ad differetiable)
More informationRiesz-Fischer Sequences and Lower Frame Bounds
Zeitschrift für Aalysis ud ihre Aweduge Joural for Aalysis ad its Applicatios Volume 1 (00), No., 305 314 Riesz-Fischer Sequeces ad Lower Frame Bouds P. Casazza, O. Christese, S. Li ad A. Lider Abstract.
More informationUniversity of Colorado Denver Dept. Math. & Stat. Sciences Applied Analysis Preliminary Exam 13 January 2012, 10:00 am 2:00 pm. Good luck!
Uiversity of Colorado Dever Dept. Math. & Stat. Scieces Applied Aalysis Prelimiary Exam 13 Jauary 01, 10:00 am :00 pm Name: The proctor will let you read the followig coditios before the exam begis, ad
More informationSTA Object Data Analysis - A List of Projects. January 18, 2018
STA 6557 Jauary 8, 208 Object Data Aalysis - A List of Projects. Schoeberg Mea glaucomatous shape chages of the Optic Nerve Head regio i aimal models 2. Aalysis of VW- Kedall ati-mea shapes with a applicatio
More informationCouncil for Innovative Research
ABSTRACT ON ABEL CONVERGENT SERIES OF FUNCTIONS ERDAL GÜL AND MEHMET ALBAYRAK Yildiz Techical Uiversity, Departmet of Mathematics, 34210 Eseler, Istabul egul34@gmail.com mehmetalbayrak12@gmail.com I this
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 informationSeries Review. a i converges if lim. i=1. a i. lim S n = lim i=1. 2 k(k + 2) converges. k=1. k=1
Defiitio: We say that the series S = Series Review i= a i is the sum of the first terms. i= a i coverges if lim S exists ad is fiite, where The above is the defiitio of covergece for series. order to see
More informationMath 113 Exam 3 Practice
Math Exam Practice Exam 4 will cover.-., 0. ad 0.. Note that eve though. was tested i exam, questios from that sectios may also be o this exam. For practice problems o., refer to the last review. This
More informationIntroduction to Extreme Value Theory Laurens de Haan, ISM Japan, Erasmus University Rotterdam, NL University of Lisbon, PT
Itroductio to Extreme Value Theory Laures de Haa, ISM Japa, 202 Itroductio to Extreme Value Theory Laures de Haa Erasmus Uiversity Rotterdam, NL Uiversity of Lisbo, PT Itroductio to Extreme Value Theory
More information6a Time change b Quadratic variation c Planar Brownian motion d Conformal local martingales e Hints to exercises...
Tel Aviv Uiversity, 28 Browia motio 59 6 Time chage 6a Time chage..................... 59 6b Quadratic variatio................. 61 6c Plaar Browia motio.............. 64 6d Coformal local martigales............
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 informations = and t = with C ij = A i B j F. (i) Note that cs = M and so ca i µ(a i ) I E (cs) = = c a i µ(a i ) = ci E (s). (ii) Note that s + t = M and so
3 From the otes we see that the parts of Theorem 4. that cocer us are: Let s ad t be two simple o-egative F-measurable fuctios o X, F, µ ad E, F F. The i I E cs ci E s for all c R, ii I E s + t I E s +
More informationProbability for mathematicians INDEPENDENCE TAU
Probability for mathematicias INDEPENDENCE TAU 2013 28 Cotets 3 Ifiite idepedet sequeces 28 3a Idepedet evets........................ 28 3b Idepedet radom variables.................. 33 3 Ifiite idepedet
More informationK. Grill Institut für Statistik und Wahrscheinlichkeitstheorie, TU Wien, Austria
MARKOV PROCESSES K. Grill Istitut für Statistik ud Wahrscheilichkeitstheorie, TU Wie, Austria Keywords: Markov process, Markov chai, Markov property, stoppig times, strog Markov property, trasitio matrix,
More informationREAL ANALYSIS II: PROBLEM SET 1 - SOLUTIONS
REAL ANALYSIS II: PROBLEM SET 1 - SOLUTIONS 18th Feb, 016 Defiitio (Lipschitz fuctio). A fuctio f : R R is said to be Lipschitz if there exists a positive real umber c such that for ay x, y i the domai
More informationDistribution 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 information2.4.2 A Theorem About Absolutely Convergent Series
0 Versio of August 27, 200 CHAPTER 2. INFINITE SERIES Add these two series: + 3 2 + 5 + 7 4 + 9 + 6 +... = 3 l 2. (2.20) 2 Sice the reciprocal of each iteger occurs exactly oce i the last series, we would
More informationTheorem 3. A subset S of a topological space X is compact if and only if every open cover of S by open sets in X has a finite subcover.
Compactess Defiitio 1. A cover or a coverig of a topological space X is a family C of subsets of X whose uio is X. A subcover of a cover C is a subfamily of C which is a cover of X. A ope cover of X is
More informationPower series are analytic
Power series are aalytic Horia Corea 1 1 The expoetial ad the logarithm For every x R we defie the fuctio give by exp(x) := 1 + x + x + + x + = x. If x = 0 we have exp(0) = 1. If x 0, cosider the series
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 informationA Note on Positive Supermartingales in Ruin Theory. Klaus D. Schmidt
A Note o Positive Supermartigales i Rui Theory Klaus D. Schmidt 94-1989 1 A ote o positive supermartigales i rui theory Klaus O. SeHMIOT Semiar für Statistik, Uiversität Maheim, A 5, 0-6800 Maheim, West
More informationThe Wasserstein distances
The Wasserstei distaces March 20, 2011 This documet presets the proof of the mai results we proved o Wasserstei distaces themselves (ad ot o curves i the Wasserstei space). I particular, triagle iequality
More informationlim za n n = z lim a n n.
Lecture 6 Sequeces ad Series Defiitio 1 By a sequece i a set A, we mea a mappig f : N A. It is customary to deote a sequece f by {s } where, s := f(). A sequece {z } of (complex) umbers is said to be coverget
More informationFinal Solutions. 1. (25pts) Define the following terms. Be as precise as you can.
Mathematics H104 A. Ogus Fall, 004 Fial Solutios 1. (5ts) Defie the followig terms. Be as recise as you ca. (a) (3ts) A ucoutable set. A ucoutable set is a set which ca ot be ut ito bijectio with a fiite
More informationSolution. 1 Solutions of Homework 1. Sangchul Lee. October 27, Problem 1.1
Solutio Sagchul Lee October 7, 017 1 Solutios of Homework 1 Problem 1.1 Let Ω,F,P) be a probability space. Show that if {A : N} F such that A := lim A exists, the PA) = lim PA ). Proof. Usig the cotiuity
More informationMath 61CM - Solutions to homework 3
Math 6CM - Solutios to homework 3 Cédric De Groote October 2 th, 208 Problem : Let F be a field, m 0 a fixed oegative iteger ad let V = {a 0 + a x + + a m x m a 0,, a m F} be the vector space cosistig
More informationON THE DELOCALIZED PHASE OF THE RANDOM PINNING MODEL
O THE DELOCALIZED PHASE OF THE RADOM PIIG MODEL JEA-CHRISTOPHE MOURRAT Abstract. We cosider the model of a directed polymer pied to a lie of i.i.d. radom charges, ad focus o the iterior of the delocalized
More informationSeries III. Chapter Alternating Series
Chapter 9 Series III With the exceptio of the Null Sequece Test, all the tests for series covergece ad divergece that we have cosidered so far have dealt oly with series of oegative terms. Series with
More information