Independent random variables
|
|
- Meagan Jacobs
- 5 years ago
- Views:
Transcription
1 CHAPTER 2 Independent random variables 2.1. Product measures Definition 2.1. Let µ i be measures on (Ω i,f i ), 1 i n. Let F F 1... F n be the sigma algebra of subsets of Ω : Ω 1... Ω n generated by all rectangles A 1... A n with A i F i. Then, the measure µ on (Ω,F ) such that µ(a 1... A n ) n i1 µ i(a i ) whenever A i F i is called a product measure and denoted µ µ 1... µ n. The existence of product measures follows along the lines of the Caratheodary construction starting with the π-system of rectangles. We skip details, but in the cases that we ever use, we shall show existence by a much neater method in Proposition 2.8. Uniqueness of product measure follows from the π λ theorem because rectangles form a π-system that generate the σ-algebra F 1... F n. Example 2.2. Let B d,m d denote the Borel sigma algebra and Lebesgue measure on R d. Then, B d B 1... B 1 and m d m 1... m 1. The first statement is clear (in fact B d+d B d B d ). Regarding m d, by definition, it is the unique measure for which m d (A 1... A n ) equals n i1 m 1(A i ) for all intervals A i. To show that it is the d-fold product of m 1, we must show that the same holds for any Borel sets A i. Fix intervals A 2,..., A n and let S : {A 1 B 1 : m d (A 1... A n ) n i1 m 1(A i )}. Then, S contains all intervals (in particular the π-system of semi-closed intervals) and by properties of measures, it is easy to check that S is a λ-system. By the π λ theorem, we get S B 1 and thus, m d (A 1... A n ) n i1 m 1(A i ) for all A 1 B 1 and any intervals A 2,..., A n. Continuing the same argument, we get that m d (A 1... A n ) n i1 m 1(A i ) for all A i B 1. The product measure property is defined in terms of sets. As always, it may be written for measurable functions and we then get the following theorem. Theorem 2.3 (Fubini s theorem). Let µ µ 1 µ 2 be a product measure on Ω 1 Ω 2 with the product σ-algebra. If f : Ω R + is either a non-negative r.v. or integrable w.r.t µ, then, (1) For every x Ω 1, the function y f (x, y) is F 2 -measurable, and the function x f (x, y)dµ 2 (y) is F 1 -measurable. The same holds with x and y interchanged. ) ) (2) f (z)dµ(z) ( f (x, y)dµ 2 (y) dµ 1 (x) ( f (x, y)dµ 1 (x) dµ 2 (y). Ω Ω 1 Ω 2 Ω 2 Ω 1 PROOF. Skipped. Attend measure theory class. Needless to day (self: then why am I saying this?) all this goes through for finite products of σ-finite measures. 25
2 26 2. INDEPENDENT RANDOM VARIABLES Infinite product measures: Given (Ω i,f i,µ i ), i 1,2,..., let Ω : Ω 1 Ω 2... and let F be the sigma algebra generated by all finite dimensional cylinders A 1... A n Ω n+1 Ω n+2... with A i F i. Does there exist a product measure µ on F? For concreteness take all (Ω i,f i,µ i ) (R,B,ν). What measure should the product measure µ give to the set A R R...? If ν(r) > 1, it is only reasonable to set µ(a R R...) to infinity, and if ν(r) < 1, it is reasonable to set it to 0. But then all cylinders will have zero measure or infinite measure!! If ν(r) 1, at least this problem does not arise. We shall show that it is indeed possible to make sense of infinite products of Thus, the only case when we can talk reasonably about infinite products of measures is for probability measures Independence Definition 2.4. Let (Ω,F,P) be a probability space. Let G 1,...,G k be sub-sigma algebras of F. We say that G i are independent if for every A 1 G 1,..., A k G k, we have P(A 1 A 2... A k ) P(A 1 )...P(A k ). Random variables X 1,..., X n on F are said to be independent if σ(x 1 ),...,σ(x n ) are independent. This is equivalent to saying that P(X i A i i k) k i1 P(X i A i ) for any A i B(R). Events A 1,..., A k are said to be independent if 1 A1,...,1 Ak are independent. This is equivalent to saying that P(A j1... A jl ) P(A j1 )...P(A jl ) for any 1 j 1 < j 2 <... < j l k. In all these cases, an infinite number of objects (sigma algebras or random variables or events) are said to be independent if every finite number of them are independent. Some remarks are in order. (1) As usual, to check independence, it would be convenient if we need check the condition in the definition only for a sufficiently large class of sets. However, if G i σ(s i ), and for every A 1 S 1,..., A k S k if we have P(A 1 A 2... A k ) P(A 1 )...P(A k ), we cannot conclude that G i are independent! If S i are π-systems, this is indeed true (see below). (2) Checking pairwise independence is insufficient to guarantee independence. For example, suppose X 1, X 2, X 3 are independent and P(X i +1) P(X i 1) 1/2. Let Y 1 X 2 X 3, Y 2 X 1 X 3 and Y 3 X 1 X 2. Then, Y i are pairwise independent but not independent. Lemma 2.5. If S i are π-systems and G i σ(s i ) and for every A 1 S 1,..., A k S k if we have P(A 1 A 2... A k ) P(A 1 )...P(A k ), then G i are independent. PROOF. Fix A 2 S 2,..., A k S k and set F 1 : {B G 1 : P(B A 2... A k ) P(B)P(A 2 )...P(A k )}. Then F 1 S 1 by assumption and it is easy to check that F 1 is a λ-system. By the π-λ theorem, it follows that F 1 G 1 and we get the assumptions of the lemma for G 1, S 2,..., S k. Repeating the argument for S 2, S 3 etc., we get independence of G 1,...,G k. Corollary 2.6. (1) Random variables X 1,..., X k are independent if and only if P(X 1 t 1,..., X k t k ) k j1 P(X j t j ). (2) Suppose G α, α I are independent. Let I 1,..., ) I k be pairwise disjoint subsets of I. Then, the σ-algebras F j σ ( α I j G α are independent.
3 2.3. INDEPENDENT SEQUENCES OF RANDOM VARIABLES 27 (3) If X i, j, i n, j n i, are independent, then for any Borel measurable f i : R n i R, the r.v.s f i (X i,1,..., X i,ni ) are also independent. PROOF. (1) The sets (, t] form a π-system that generates B(R). (2) For j k, let S j be the collection of finite intersections of sets A i, i I j. Then S j are π-systems and σ(s j ) F j. (3) Follows from (2) by considering G i, j : σ(x i, j ) and observing that f i (X i,1,..., X i,k ) σ(g i,1... G i,ni ). So far, we stated conditions for independence in terms of probabilities if events. As usual, they generalize to conditions in terms of expectations of random variables. Lemma 2.7. (1) Sigma algebras G 1,...,G k are independent if and only if for every bounded G i -measurable functions X i, 1 i k, we have, E[X 1... X k ] k i1 E[X i]. (2) In particular, random variables Z 1,..., Z k (Z i is an n i dimensional random vector) are independent if and only if E[ k i1 f i(z i )] k i1 E[f i(z i )] for any bounded Borel measurable functions f i : R n i R. We say bounded measurable just to ensure that expectations exist. The proof goes inductively by fixing X 2,..., X k and then letting X 1 be a simple r.v., a nonnegative r.v. and a general bounded measurable r.v. PROOF. (1) Suppose G i are independent. If X i are G i measurable then it is clear that X i are independent and hence P(X 1,..., X k ) 1 PX PX 1 k. Denote µ i : PX 1 and apply Fubini s theorem (and change of i variables) to get E[X 1... X k ] c.o.v Fub R k i1 R x i d(µ 1... µ k )(x 1,..., x k )... x i dµ 1 (x 1 )... dµ k (x k ) i1 R i1 udµ i (u) c.o.v R E[X i ]. Conversely, if E[X 1... X k ] k i1 E[X i] for all G i -measurable functions X i s, then applying to indicators of events A i G i we see the independence of the σ-algebras G i. (2) The second claim follows from the first by setting G i : σ(x i ) and observing that a random variable X i is σ(z i )-measurable if and only if X f Z i for some Borel measurable f : R n i R. i Independent sequences of random variables First we make the observation that product measures and independence are closely related concepts. For example, An observation: The independence of random variables X 1,..., X k is precisely the same as saying that P X 1 is the product measure PX PXk, where X (X 1,..., X k ).
4 28 2. INDEPENDENT RANDOM VARIABLES Consider the following questions. Henceforth, we write R for the countable product space R R... and B(R ) for the cylinder σ-algebra generated by all finite dimensional cylinders A 1... A n R R... with A i B(R). This notation is justified, becaue the cylinder σ-algebra is also the Borel σ-algebra on R with the product topology. Question 1: Given µ i P (R), i 1, does there exist a probability space with independent random variables X i having distributions µ i? Question 2: Given µ i P (R), i 1, does there exist a p.m µ on (R,B(R )) such that µ(a 1... A n R R...) n i1 µ i(a i )? Observation: The above two questions are equivalent. For, suppose we answer the first question by finding an (Ω,F,P) with independent random variables X i : Ω R such that X i µ i for all i. Then, X : Ω R defined by X(ω) (X 1 (ω), X 2 (ω),...) is measurable w.r.t the relevant σ-algebras (why?). Then, let µ : PX 1 be the pushforward p.m on R. Clearly µ(a 1... A n R R...) P(X 1 A 1,..., X n A n ) n n P(X i A i ) µ i (A i ). Thus µ is the product measure required by the second question. Conversely, if we could construct the product measure on (R,B(R )), then we could take Ω R, F B(R ) and X i to be the i th co-ordinate random variable. Then you may check that they satisfy the requirements of the first question. The two questions are thus equivalent, but what is the answer?! It is yes, of course or we would not make heavy weather about it. Proposition 2.8 (Daniell). Let µ i P (R), i 1, be Borel p.m on R. Then, there exist a probability space with independent random variables X 1, X 2,... such that X i µ i. PROOF. We arrive at the construction in three stages. (1) Independent Bernoullis: Consider ([0, 1], B, m) and the random variables X k : [0,1] R, where X k (ω) is defined to be the k th digit in the binary expansion of ω. For definiteness, we may always take the infinite binary expansion. Then by an earlier homework exercise, X 1, X 2,... are independent Bernoulli(1/2) random variables. (2) Independent uniforms: Note that as a consequence, on any probability space, if Y i are i.i.d. Ber(1/2) variables, then U : n1 2 n Y n has uniform distribution on [0, 1]. Consider again the canonical probability space and the r.v. X i, and set U 1 : X 1 /2 + X 3 /2 3 + X 5 / , U 2 : X 2 /2 + X 6 / , etc. Clearly, U i are i.i.d. U[0,1]. (3) Arbitrary distributions: For a p.m. µ, recall the left-continuous inverse G µ that had the property that G µ (U) µ if U U[0,1]. Suppose we are given p.m.s µ 1,µ 2,... On the canonical probability space, let U i be i.i.d uniforms constructed as before. Define X i : G µi (U i ). Then, X i are independent and X i µ i. Thus we have constructed an independent sequence of random variables having the specified distributions. i1 i1 Sometimes in books one finds construction of uncountable product measures too. It has no use. But a very natural question at this point is to go beyond independence. We just state the following theorem which generalizes the previous proposition.
5 2.3. INDEPENDENT SEQUENCES OF RANDOM VARIABLES 29 Theorem 2.9 (Kolmogorov s existence theorem). For each n 1 and each 1 i 1 < i 2 <... < i n, let µ i1,...,i n be a Borel p.m on R n. Then there exists a unique probability measure µ on (R,B(R )) such that µ(a 1... A n R R...) µ i1,...,i n (A 1... A n ) for all n 1 and all A i B(R), if and only if the given family of probability measures satisfy the consistency condition µ i1,...,i n (A 1... A n 1 R) µ i1,...,i n 1 (A 1... A n 1 ) for any A k B(R) and for any 1 i 1 < i 2 <... < i n and any n 1.
Section 2: Classes of Sets
Section 2: Classes of Sets Notation: If A, B are subsets of X, then A \ B denotes the set difference, A \ B = {x A : x B}. A B denotes the symmetric difference. A B = (A \ B) (B \ A) = (A B) \ (A B). Remarks
More informationSTA 711: Probability & Measure Theory Robert L. Wolpert
STA 711: Probability & Measure Theory Robert L. Wolpert 6 Independence 6.1 Independent Events A collection of events {A i } F in a probability space (Ω,F,P) is called independent if P[ i I A i ] = P[A
More informationStat 451: Solutions to Assignment #1
Stat 451: Solutions to Assignment #1 2.1) By definition, 2 Ω is the set of all subsets of Ω. Therefore, to show that 2 Ω is a σ-algebra we must show that the conditions of the definition σ-algebra are
More informationAbstract Measure Theory
2 Abstract Measure Theory Lebesgue measure is one of the premier examples of a measure on R d, but it is not the only measure and certainly not the only important measure on R d. Further, R d is not the
More informationProduct measures, Tonelli s and Fubini s theorems For use in MAT4410, autumn 2017 Nadia S. Larsen. 17 November 2017.
Product measures, Tonelli s and Fubini s theorems For use in MAT4410, autumn 017 Nadia S. Larsen 17 November 017. 1. Construction of the product measure The purpose of these notes is to prove the main
More information18.175: Lecture 2 Extension theorems, random variables, distributions
18.175: Lecture 2 Extension theorems, random variables, distributions Scott Sheffield MIT Outline Extension theorems Characterizing measures on R d Random variables Outline Extension theorems Characterizing
More informationSpring 2014 Advanced Probability Overview. Lecture Notes Set 1: Course Overview, σ-fields, and Measures
36-752 Spring 2014 Advanced Probability Overview Lecture Notes Set 1: Course Overview, σ-fields, and Measures Instructor: Jing Lei Associated reading: Sec 1.1-1.4 of Ash and Doléans-Dade; Sec 1.1 and A.1
More informationThe Kolmogorov extension theorem
The Kolmogorov extension theorem Jordan Bell jordan.bell@gmail.com Department of Mathematics, University of Toronto June 21, 2014 1 σ-algebras and semirings If X is a nonempty set, an algebra of sets on
More informationBuilding Infinite Processes from Finite-Dimensional Distributions
Chapter 2 Building Infinite Processes from Finite-Dimensional Distributions Section 2.1 introduces the finite-dimensional distributions of a stochastic process, and shows how they determine its infinite-dimensional
More informationLecture 7. Sums of random variables
18.175: Lecture 7 Sums of random variables Scott Sheffield MIT 18.175 Lecture 7 1 Outline Definitions Sums of random variables 18.175 Lecture 7 2 Outline Definitions Sums of random variables 18.175 Lecture
More information13. Examples of measure-preserving tranformations: rotations of a torus, the doubling map
3. Examples of measure-preserving tranformations: rotations of a torus, the doubling map 3. Rotations of a torus, the doubling map In this lecture we give two methods by which one can show that a given
More informationWe will briefly look at the definition of a probability space, probability measures, conditional probability and independence of probability events.
1 Probability 1.1 Probability spaces We will briefly look at the definition of a probability space, probability measures, conditional probability and independence of probability events. Definition 1.1.
More informationREAL ANALYSIS I Spring 2016 Product Measures
REAL ANALSIS I Spring 216 Product Measures We assume that (, M, µ), (, N, ν) are σ- finite measure spaces. We want to provide the Cartesian product with a measure space structure in which all sets of the
More informationProbability and Measure
Part II Year 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2018 84 Paper 4, Section II 26J Let (X, A) be a measurable space. Let T : X X be a measurable map, and µ a probability
More informationMATH 418: Lectures on Conditional Expectation
MATH 418: Lectures on Conditional Expectation Instructor: r. Ed Perkins, Notes taken by Adrian She Conditional expectation is one of the most useful tools of probability. The Radon-Nikodym theorem enables
More informationMeasure and Integration: Concepts, Examples and Exercises. INDER K. RANA Indian Institute of Technology Bombay India
Measure and Integration: Concepts, Examples and Exercises INDER K. RANA Indian Institute of Technology Bombay India Department of Mathematics, Indian Institute of Technology, Bombay, Powai, Mumbai 400076,
More informationMeasures and Measure Spaces
Chapter 2 Measures and Measure Spaces In summarizing the flaws of the Riemann integral we can focus on two main points: 1) Many nice functions are not Riemann integrable. 2) The Riemann integral does not
More informationErgodic Theorems. Samy Tindel. Purdue University. Probability Theory 2 - MA 539. Taken from Probability: Theory and examples by R.
Ergodic Theorems Samy Tindel Purdue University Probability Theory 2 - MA 539 Taken from Probability: Theory and examples by R. Durrett Samy T. Ergodic theorems Probability Theory 1 / 92 Outline 1 Definitions
More informationIn terms of measures: Exercise 1. Existence of a Gaussian process: Theorem 2. Remark 3.
1. GAUSSIAN PROCESSES A Gaussian process on a set T is a collection of random variables X =(X t ) t T on a common probability space such that for any n 1 and any t 1,...,t n T, the vector (X(t 1 ),...,X(t
More informationAnnalee Gomm Math 714: Assignment #2
Annalee Gomm Math 714: Assignment #2 3.32. Verify that if A M, λ(a = 0, and B A, then B M and λ(b = 0. Suppose that A M with λ(a = 0, and let B be any subset of A. By the nonnegativity and monotonicity
More informationAdmin and Lecture 1: Recap of Measure Theory
Admin and Lecture 1: Recap of Measure Theory David Aldous January 16, 2018 I don t use bcourses: Read web page (search Aldous 205B) Web page rather unorganized some topics done by Nike in 205A will post
More informationRS Chapter 1 Random Variables 6/5/2017. Chapter 1. Probability Theory: Introduction
Chapter 1 Probability Theory: Introduction Basic Probability General In a probability space (Ω, Σ, P), the set Ω is the set of all possible outcomes of a probability experiment. Mathematically, Ω is just
More informationMeasure and integration
Chapter 5 Measure and integration In calculus you have learned how to calculate the size of different kinds of sets: the length of a curve, the area of a region or a surface, the volume or mass of a solid.
More informationµ (X) := inf l(i k ) where X k=1 I k, I k an open interval Notice that is a map from subsets of R to non-negative number together with infinity
A crash course in Lebesgue measure theory, Math 317, Intro to Analysis II These lecture notes are inspired by the third edition of Royden s Real analysis. The Jordan content is an attempt to extend the
More informationProbability Theory. Richard F. Bass
Probability Theory Richard F. Bass ii c Copyright 2014 Richard F. Bass Contents 1 Basic notions 1 1.1 A few definitions from measure theory............. 1 1.2 Definitions............................. 2
More informationII - REAL ANALYSIS. This property gives us a way to extend the notion of content to finite unions of rectangles: we define
1 Measures 1.1 Jordan content in R N II - REAL ANALYSIS Let I be an interval in R. Then its 1-content is defined as c 1 (I) := b a if I is bounded with endpoints a, b. If I is unbounded, we define c 1
More informationChapter 1. Measure Spaces. 1.1 Algebras and σ algebras of sets Notation and preliminaries
Chapter 1 Measure Spaces 1.1 Algebras and σ algebras of sets 1.1.1 Notation and preliminaries We shall denote by X a nonempty set, by P(X) the set of all parts (i.e., subsets) of X, and by the empty set.
More informationDynkin (λ-) and π-systems; monotone classes of sets, and of functions with some examples of application (mainly of a probabilistic flavor)
Dynkin (λ-) and π-systems; monotone classes of sets, and of functions with some examples of application (mainly of a probabilistic flavor) Matija Vidmar February 7, 2018 1 Dynkin and π-systems Some basic
More information1 Measurable Functions
36-752 Advanced Probability Overview Spring 2018 2. Measurable Functions, Random Variables, and Integration Instructor: Alessandro Rinaldo Associated reading: Sec 1.5 of Ash and Doléans-Dade; Sec 1.3 and
More informationExercises Measure Theoretic Probability
Exercises Measure Theoretic Probability 2002-2003 Week 1 1. Prove the folloing statements. (a) The intersection of an arbitrary family of d-systems is again a d- system. (b) The intersection of an arbitrary
More informationConstruction of a general measure structure
Chapter 4 Construction of a general measure structure We turn to the development of general measure theory. The ingredients are a set describing the universe of points, a class of measurable subsets along
More information(U) =, if 0 U, 1 U, (U) = X, if 0 U, and 1 U. (U) = E, if 0 U, but 1 U. (U) = X \ E if 0 U, but 1 U. n=1 A n, then A M.
1. Abstract Integration The main reference for this section is Rudin s Real and Complex Analysis. The purpose of developing an abstract theory of integration is to emphasize the difference between the
More information1.1. MEASURES AND INTEGRALS
CHAPTER 1: MEASURE THEORY In this chapter we define the notion of measure µ on a space, construct integrals on this space, and establish their basic properties under limits. The measure µ(e) will be defined
More informationLebesgue measure on R is just one of many important measures in mathematics. In these notes we introduce the general framework for measures.
Measures In General Lebesgue measure on R is just one of many important measures in mathematics. In these notes we introduce the general framework for measures. Definition: σ-algebra Let X be a set. A
More informationMATH41011/MATH61011: FOURIER SERIES AND LEBESGUE INTEGRATION. Extra Reading Material for Level 4 and Level 6
MATH41011/MATH61011: FOURIER SERIES AND LEBESGUE INTEGRATION Extra Reading Material for Level 4 and Level 6 Part A: Construction of Lebesgue Measure The first part the extra material consists of the construction
More information1 Sequences of events and their limits
O.H. Probability II (MATH 2647 M15 1 Sequences of events and their limits 1.1 Monotone sequences of events Sequences of events arise naturally when a probabilistic experiment is repeated many times. For
More informationHILBERT SPACES AND THE RADON-NIKODYM THEOREM. where the bar in the first equation denotes complex conjugation. In either case, for any x V define
HILBERT SPACES AND THE RADON-NIKODYM THEOREM STEVEN P. LALLEY 1. DEFINITIONS Definition 1. A real inner product space is a real vector space V together with a symmetric, bilinear, positive-definite mapping,
More informationChapter 4. Measure Theory. 1. Measure Spaces
Chapter 4. Measure Theory 1. Measure Spaces Let X be a nonempty set. A collection S of subsets of X is said to be an algebra on X if S has the following properties: 1. X S; 2. if A S, then A c S; 3. if
More informationAdvanced Probability
Advanced Probability Perla Sousi October 10, 2011 Contents 1 Conditional expectation 1 1.1 Discrete case.................................. 3 1.2 Existence and uniqueness............................ 3 1
More informationNotes on Measure, Probability and Stochastic Processes. João Lopes Dias
Notes on Measure, Probability and Stochastic Processes João Lopes Dias Departamento de Matemática, ISEG, Universidade de Lisboa, Rua do Quelhas 6, 1200-781 Lisboa, Portugal E-mail address: jldias@iseg.ulisboa.pt
More information1.4 Outer measures 10 CHAPTER 1. MEASURE
10 CHAPTER 1. MEASURE 1.3.6. ( Almost everywhere and null sets If (X, A, µ is a measure space, then a set in A is called a null set (or µ-null if its measure is 0. Clearly a countable union of null sets
More information36-752: Lecture 1. We will use measures to say how large sets are. First, we have to decide which sets we will measure.
0 0 0 -: Lecture How is this course different from your earlier probability courses? There are some problems that simply can t be handled with finite-dimensional sample spaces and random variables that
More informationQuick review on Discrete Random Variables
STAT/MATH 395 A - PROBABILITY II UW Winter Quarter 2017 Néhémy Lim Quick review on Discrete Random Variables Notations. Z = {..., 2, 1, 0, 1, 2,...}, set of all integers; N = {0, 1, 2,...}, set of natural
More informationRandom Process Lecture 1. Fundamentals of Probability
Random Process Lecture 1. Fundamentals of Probability Husheng Li Min Kao Department of Electrical Engineering and Computer Science University of Tennessee, Knoxville Spring, 2016 1/43 Outline 2/43 1 Syllabus
More informationBasic Measure and Integration Theory. Michael L. Carroll
Basic Measure and Integration Theory Michael L. Carroll Sep 22, 2002 Measure Theory: Introduction What is measure theory? Why bother to learn measure theory? 1 What is measure theory? Measure theory is
More information18.175: Lecture 3 Integration
18.175: Lecture 3 Scott Sheffield MIT Outline Outline Recall definitions Probability space is triple (Ω, F, P) where Ω is sample space, F is set of events (the σ-algebra) and P : F [0, 1] is the probability
More information1. Probability Measure and Integration Theory in a Nutshell
1. Probability Measure and Integration Theory in a Nutshell 1.1. Measurable Space and Measurable Functions Definition 1.1. A measurable space is a tuple (Ω, F) where Ω is a set and F a σ-algebra on Ω,
More informationNotes 1 : Measure-theoretic foundations I
Notes 1 : Measure-theoretic foundations I Math 733-734: Theory of Probability Lecturer: Sebastien Roch References: [Wil91, Section 1.0-1.8, 2.1-2.3, 3.1-3.11], [Fel68, Sections 7.2, 8.1, 9.6], [Dur10,
More information4. Ergodicity and mixing
4. Ergodicity and mixing 4. Introduction In the previous lecture we defined what is meant by an invariant measure. In this lecture, we define what is meant by an ergodic measure. The primary motivation
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 3 9/10/2008 CONDITIONING AND INDEPENDENCE
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 3 9/10/2008 CONDITIONING AND INDEPENDENCE Most of the material in this lecture is covered in [Bertsekas & Tsitsiklis] Sections 1.3-1.5
More informationMeasures. 1 Introduction. These preliminary lecture notes are partly based on textbooks by Athreya and Lahiri, Capinski and Kopp, and Folland.
Measures These preliminary lecture notes are partly based on textbooks by Athreya and Lahiri, Capinski and Kopp, and Folland. 1 Introduction Our motivation for studying measure theory is to lay a foundation
More informationLebesgue Measure on R n
CHAPTER 2 Lebesgue Measure on R n Our goal is to construct a notion of the volume, or Lebesgue measure, of rather general subsets of R n that reduces to the usual volume of elementary geometrical sets
More informationLecture Notes 1 Probability and Random Variables. Conditional Probability and Independence. Functions of a Random Variable
Lecture Notes 1 Probability and Random Variables Probability Spaces Conditional Probability and Independence Random Variables Functions of a Random Variable Generation of a Random Variable Jointly Distributed
More information1 Probability space and random variables
1 Probability space and random variables As graduate level, we inevitably need to study probability based on measure theory. It obscures some intuitions in probability, but it also supplements our intuition,
More informationExercises for Unit VI (Infinite constructions in set theory)
Exercises for Unit VI (Infinite constructions in set theory) VI.1 : Indexed families and set theoretic operations (Halmos, 4, 8 9; Lipschutz, 5.3 5.4) Lipschutz : 5.3 5.6, 5.29 5.32, 9.14 1. Generalize
More informationCHANGE OF MEASURE. D.Majumdar
CHANGE OF MEASURE D.Majumdar We had touched upon this concept when we looked at Finite Probability spaces and had defined a R.V. Z to change probability measure on a space Ω. We need to do the same thing
More informationLecture 4: Conditional expectation and independence
Lecture 4: Conditional expectation and independence In elementry probability, conditional probability P(B A) is defined as P(B A) P(A B)/P(A) for events A and B with P(A) > 0. For two random variables,
More informationReal Variables: Solutions to Homework 3
Real Variables: Solutions to Homework 3 September 3, 011 Exercise 0.1. Chapter 3, # : Show that the cantor set C consists of all x such that x has some triadic expansion for which every is either 0 or.
More informationµ X (A) = P ( X 1 (A) )
1 STOCHASTIC PROCESSES This appendix provides a very basic introduction to the language of probability theory and stochastic processes. We assume the reader is familiar with the general measure and integration
More information4th Preparation Sheet - Solutions
Prof. Dr. Rainer Dahlhaus Probability Theory Summer term 017 4th Preparation Sheet - Solutions Remark: Throughout the exercise sheet we use the two equivalent definitions of separability of a metric space
More informationMathematics Course 111: Algebra I Part I: Algebraic Structures, Sets and Permutations
Mathematics Course 111: Algebra I Part I: Algebraic Structures, Sets and Permutations D. R. Wilkins Academic Year 1996-7 1 Number Systems and Matrix Algebra Integers The whole numbers 0, ±1, ±2, ±3, ±4,...
More informationMeasure Theory on Topological Spaces. Course: Prof. Tony Dorlas 2010 Typset: Cathal Ormond
Measure Theory on Topological Spaces Course: Prof. Tony Dorlas 2010 Typset: Cathal Ormond May 22, 2011 Contents 1 Introduction 2 1.1 The Riemann Integral........................................ 2 1.2 Measurable..............................................
More information2 Construction & Extension of Measures
STA 711: Probability & Measure Theory Robert L. Wolpert 2 Construction & Extension of Measures For any finite set Ω = {ω 1,...,ω n }, the power set PΩ is the collection of all subsets of Ω, including the
More information3 (Due ). Let A X consist of points (x, y) such that either x or y is a rational number. Is A measurable? What is its Lebesgue measure?
MA 645-4A (Real Analysis), Dr. Chernov Homework assignment 1 (Due ). Show that the open disk x 2 + y 2 < 1 is a countable union of planar elementary sets. Show that the closed disk x 2 + y 2 1 is a countable
More informationStochastic Processes. Winter Term Paolo Di Tella Technische Universität Dresden Institut für Stochastik
Stochastic Processes Winter Term 2016-2017 Paolo Di Tella Technische Universität Dresden Institut für Stochastik Contents 1 Preliminaries 5 1.1 Uniform integrability.............................. 5 1.2
More informationHints/Solutions for Homework 3
Hints/Solutions for Homework 3 MATH 865 Fall 25 Q Let g : and h : be bounded and non-decreasing functions Prove that, for any rv X, [Hint: consider an independent copy Y of X] ov(g(x), h(x)) Solution:
More informationMATH31011/MATH41011/MATH61011: FOURIER ANALYSIS AND LEBESGUE INTEGRATION. Chapter 2: Countability and Cantor Sets
MATH31011/MATH41011/MATH61011: FOURIER ANALYSIS AND LEBESGUE INTEGRATION Chapter 2: Countability and Cantor Sets Countable and Uncountable Sets The concept of countability will be important in this course
More informationContinuum Probability and Sets of Measure Zero
Chapter 3 Continuum Probability and Sets of Measure Zero In this chapter, we provide a motivation for using measure theory as a foundation for probability. It uses the example of random coin tossing to
More informationLebesgue Measure. Dung Le 1
Lebesgue Measure Dung Le 1 1 Introduction How do we measure the size of a set in IR? Let s start with the simplest ones: intervals. Obviously, the natural candidate for a measure of an interval is its
More informationProbability and Measure. November 27, 2017
Probability and Measure November 27, 2017 1 CONTENTS 2 Contents 1 Measure Theory 4 1.1 History................................ 4 1.2 Lebesgue Measure.......................... 5 1.3 Measurable Functions........................
More informationChapter 1: Probability Theory Lecture 1: Measure space, measurable function, and integration
Chapter 1: Probability Theory Lecture 1: Measure space, measurable function, and integration Random experiment: uncertainty in outcomes Ω: sample space: a set containing all possible outcomes Definition
More informationReminder Notes for the Course on Measures on Topological Spaces
Reminder Notes for the Course on Measures on Topological Spaces T. C. Dorlas Dublin Institute for Advanced Studies School of Theoretical Physics 10 Burlington Road, Dublin 4, Ireland. Email: dorlas@stp.dias.ie
More informationCW complexes. Soren Hansen. This note is meant to give a short introduction to CW complexes.
CW complexes Soren Hansen This note is meant to give a short introduction to CW complexes. 1. Notation and conventions In the following a space is a topological space and a map f : X Y between topological
More informationLecture 17 Brownian motion as a Markov process
Lecture 17: Brownian motion as a Markov process 1 of 14 Course: Theory of Probability II Term: Spring 2015 Instructor: Gordan Zitkovic Lecture 17 Brownian motion as a Markov process Brownian motion is
More informationLecture Notes 1 Probability and Random Variables. Conditional Probability and Independence. Functions of a Random Variable
Lecture Notes 1 Probability and Random Variables Probability Spaces Conditional Probability and Independence Random Variables Functions of a Random Variable Generation of a Random Variable Jointly Distributed
More informationRandom experiments may consist of stages that are performed. Example: Roll a die two times. Consider the events E 1 = 1 or 2 on first roll
Econ 514: Probability and Statistics Lecture 4: Independence Stochastic independence Random experiments may consist of stages that are performed independently. Example: Roll a die two times. Consider the
More informationThree hours THE UNIVERSITY OF MANCHESTER. 31st May :00 17:00
Three hours MATH41112 THE UNIVERSITY OF MANCHESTER ERGODIC THEORY 31st May 2016 14:00 17:00 Answer FOUR of the FIVE questions. If more than four questions are attempted, then credit will be given for the
More informationMATH/STAT 235A Probability Theory Lecture Notes, Fall 2013
MATH/STAT 235A Probability Theory Lecture Notes, Fall 2013 Dan Romik Department of Mathematics, UC Davis December 30, 2013 Contents Chapter 1: Introduction 6 1.1 What is probability theory?...........................
More informationMATHS 730 FC Lecture Notes March 5, Introduction
1 INTRODUCTION MATHS 730 FC Lecture Notes March 5, 2014 1 Introduction Definition. If A, B are sets and there exists a bijection A B, they have the same cardinality, which we write as A, #A. If there exists
More informationconsists of two disjoint copies of X n, each scaled down by 1,
Homework 4 Solutions, Real Analysis I, Fall, 200. (4) Let be a topological space and M be a σ-algebra on which contains all Borel sets. Let m, µ be two positive measures on M. Assume there is a constant
More information5 Measure theory II. (or. lim. Prove the proposition. 5. For fixed F A and φ M define the restriction of φ on F by writing.
5 Measure theory II 1. Charges (signed measures). Let (Ω, A) be a σ -algebra. A map φ: A R is called a charge, (or signed measure or σ -additive set function) if φ = φ(a j ) (5.1) A j for any disjoint
More informationLecture 11: Random Variables
EE5110: Probability Foundations for Electrical Engineers July-November 2015 Lecture 11: Random Variables Lecturer: Dr. Krishna Jagannathan Scribe: Sudharsan, Gopal, Arjun B, Debayani The study of random
More informationProbability and Random Processes
Probability and Random Processes Lecture 7 Conditional probability and expectation Decomposition of measures Mikael Skoglund, Probability and random processes 1/13 Conditional Probability A probability
More informationIntegral Jensen inequality
Integral Jensen inequality Let us consider a convex set R d, and a convex function f : (, + ]. For any x,..., x n and λ,..., λ n with n λ i =, we have () f( n λ ix i ) n λ if(x i ). For a R d, let δ a
More informationLecture 9. d N(0, 1). Now we fix n and think of a SRW on [0,1]. We take the k th step at time k n. and our increments are ± 1
Random Walks and Brownian Motion Tel Aviv University Spring 011 Lecture date: May 0, 011 Lecture 9 Instructor: Ron Peled Scribe: Jonathan Hermon In today s lecture we present the Brownian motion (BM).
More information2 Probability, random elements, random sets
Tel Aviv University, 2012 Measurability and continuity 25 2 Probability, random elements, random sets 2a Probability space, measure algebra........ 25 2b Standard models................... 30 2c Random
More information2 (Bonus). Let A X consist of points (x, y) such that either x or y is a rational number. Is A measurable? What is its Lebesgue measure?
MA 645-4A (Real Analysis), Dr. Chernov Homework assignment 1 (Due 9/5). Prove that every countable set A is measurable and µ(a) = 0. 2 (Bonus). Let A consist of points (x, y) such that either x or y is
More informationProbability and Measure
Probability and Measure Martin Orr 19th January 2009 0.1 Introduction This is a set of notes for the Probability and Measure course in Part II of the Cambridge Mathematics Tripos. They were written as
More informationBrownian Motion and Conditional Probability
Math 561: Theory of Probability (Spring 2018) Week 10 Brownian Motion and Conditional Probability 10.1 Standard Brownian Motion (SBM) Brownian motion is a stochastic process with both practical and theoretical
More informationCompendium and Solutions to exercises TMA4225 Foundation of analysis
Compendium and Solutions to exercises TMA4225 Foundation of analysis Ruben Spaans December 6, 2010 1 Introduction This compendium contains a lexicon over definitions and exercises with solutions. Throughout
More informationAn Introduction to Entropy and Subshifts of. Finite Type
An Introduction to Entropy and Subshifts of Finite Type Abby Pekoske Department of Mathematics Oregon State University pekoskea@math.oregonstate.edu August 4, 2015 Abstract This work gives an overview
More informationExercises Measure Theoretic Probability
Exercises Measure Theoretic Probability Chapter 1 1. Prove the folloing statements. (a) The intersection of an arbitrary family of d-systems is again a d- system. (b) The intersection of an arbitrary family
More informationMeasure Theoretic Probability. P.J.C. Spreij
Measure Theoretic Probability P.J.C. Spreij this version: September 16, 2009 Contents 1 σ-algebras and measures 1 1.1 σ-algebras............................... 1 1.2 Measures...............................
More informationFinal. due May 8, 2012
Final due May 8, 2012 Write your solutions clearly in complete sentences. All notation used must be properly introduced. Your arguments besides being correct should be also complete. Pay close attention
More informationIndeed, if we want m to be compatible with taking limits, it should be countably additive, meaning that ( )
Lebesgue Measure The idea of the Lebesgue integral is to first define a measure on subsets of R. That is, we wish to assign a number m(s to each subset S of R, representing the total length that S takes
More informationFE 5204 Stochastic Differential Equations
Instructor: Jim Zhu e-mail:zhu@wmich.edu http://homepages.wmich.edu/ zhu/ January 20, 2009 Preliminaries for dealing with continuous random processes. Brownian motions. Our main reference for this lecture
More informationTHEOREMS, ETC., FOR MATH 516
THEOREMS, ETC., FOR MATH 516 Results labeled Theorem Ea.b.c (or Proposition Ea.b.c, etc.) refer to Theorem c from section a.b of Evans book (Partial Differential Equations). Proposition 1 (=Proposition
More informationBasic Definitions: Indexed Collections and Random Functions
Chapter 1 Basic Definitions: Indexed Collections and Random Functions Section 1.1 introduces stochastic processes as indexed collections of random variables. Section 1.2 builds the necessary machinery
More informationLecture 6 Basic Probability
Lecture 6: Basic Probability 1 of 17 Course: Theory of Probability I Term: Fall 2013 Instructor: Gordan Zitkovic Lecture 6 Basic Probability Probability spaces A mathematical setup behind a probabilistic
More informationIntroduction to Ergodic Theory
Lecture I Crash course in measure theory Oliver Butterley, Irene Pasquinelli, Stefano Luzzatto, Lucia Simonelli, Davide Ravotti Summer School in Dynamics ICTP 2018 Why do we care about measure theory?
More information