Transcription of Math 280 (Probability Theory) Lecture Notes
1 Bruce K. DriverMath 280 ( probability theory ) Lecture NotesMarch 12, 2007 Homework Problems:-1 math 280B Homework Problems.. Homework 1. Due Monday, January 22, 2007 .. Homework 2. Due Monday, January 29, 2007 .. Homework #3 Due Monday, February 5, 2007 .. Homework #4 Due Friday, February 16, 2007 .. Homework #5 Due Friday, February 23, 2007 .. Homework #6 Due Monday, March 5, 2007 .. Homework #7 Due Monday, March 12, 2007 .. Homework #8 Due Monday, March 19, 2007 .. 40 math 280A Homework Problems.. Homework 1. Due Friday, September 29, 2006 .. Homework 2. Due Friday, October 6, 2006 .. Homework 3. Due Friday, October 13, 2006 .. Homework 4. Due Friday, October 20, 2006 .. Homework 5. Due Friday, October 27, 2006 .. Homework 6. Due Friday, November 3, 2006 .. Homework 7. Due Monday, November 13, 2006.
2 Corrections and comments on Homework 7 (280A) .. Homework 8. Due Monday, November 27, 2006 .. Homework 9. Due Noon, on Wednesday, December 6, 2006 .. 6 Part I Background Material1 Limsups, Liminfs and Extended Limits..94 Contents2 Basic Probabilistic Notions..13 Part II Formal Development3 Preliminaries.. Set Operations .. Exercises .. Algebraic sub-structures of sets .. 214 Finitely Additive Measures.. Finitely Additive Measures .. Examples of Measures.. Simple Integration .. Simple Independence and the Weak Law of Large Numbers .. Constructing Finitely Additive Measures .. 325 Countably Additive Measures.. Distribution Function for probability Measures on (R,BR) .. Construction of Premeasures .. Regularity and Uniqueness Results .. Construction of Measures .. Completions of Measure Spaces.
3 A Baby Version of Kolmogorov s Extension Theorem.. 426 Random Variables.. Measurable Functions .. Factoring Random Variables .. 497 Independence.. and Monotone Class Theorems .. The Monotone Class Theorem .. Basic Properties of Independence .. An Example of Ranks .. Borel-Cantelli Lemmas .. Kolmogorov and Hewitt-Savage Zero-One Laws.. 638 Integration theory .. A Quick Introduction to Lebesgue Integration theory .. Integrals of positive functions .. Integrals of Complex Valued Functions .. Densities and Change of Variables Theorems .. Measurability on Complete Measure Spaces .. 81 Page: 4 job: prob macro: date/time: 12-Mar-2007/12 Comparison of the Lebesgue and the Riemann Integral .. Exercises.
4 Laws of Large Numbers Exercises .. 849 Functional Forms of the Theorem..8510 Multiple and Iterated Integrals.. Iterated Integrals .. Tonelli s Theorem and Product Measure .. Fubini s Theorem .. Fubini s Theorem and Completions .. Lebesgue Measure onRdand the Change of Variables Theorem .. The Polar Decomposition of Lebesgue Measure .. More Spherical Coordinates .. Exercises .. 10611Lp spaces.. Modes of Convergence .. Jensen s, H older s and Minikowski s Inequalities .. Completeness ofLp spaces .. Relationships between differentLp spaces .. Summary: .. Uniform Integrability .. Exercises .. Appendix: Convex Functions .. 119 Part III Convergence Results12 Laws of Large Numbers.. Main Results .. Examples.. Random Series Examples .. A WLLN Example .. Strong Law of Large Number Examples.
5 More on the Weak Laws of Large Numbers .. Maximal Inequalities .. Kolmogorov s Convergence Criteria and the SSLN .. Strong Law of Large Numbers.. Necessity Proof of Kolmogorov s Three Series Theorem .. 140 Page: 5 job: prob macro: date/time: 12-Mar-2007/12:256 Contents13 Weak Convergence Results.. Total Variation Distance .. Weak Convergence.. Derived Weak Convergence .. Skorohod and the Convergence of Types Theorems .. Weak Convergence Examples .. Compactness and Tightness.. Weak Convergence in Metric Spaces .. 15714 Characteristic Functions (Fourier Transform).. Basic Properties of the Characteristic Function .. Examples.. Continuity Theorem .. A Fourier Transform Inversion Formula .. Exercises .. Appendix: Bochner s Theorem.
6 Appendix: A Multi-dimensional Weirstrass Approximation Theorem .. Appendix: Some Calculus Estimates .. 17715 Weak Convergence of Random Sums.. Infinitely Divisible and Stable Symmetric Distributions .. Stable Laws .. 189 Part IV Conditional Expectations and Martingales16 Hilbert Space Basics.. Compactness Results forLp Spaces .. Exercises .. 19917 The Radon-Nikodym Theorem..20118 Conditional Expectation.. Examples.. Additional Properties of Conditional Expectations .. Regular Conditional Distributions .. Appendix: Standard Borel Spaces.. 21319 (Sub and Super) Martingales.. (Sub and Super) Martingale Examples .. Decompositions .. Stopping Times .. Stochastic Integrals, Optional Stopping, and Switching .. 224 Page: 6 job: prob macro: date/time: 12-Mar-2007/12 Maximal Inequalities.
7 Upcrossing Inequalities and Convergence Results .. Closure and Regularity Results .. 232 References..235 Page: 7 job: prob macro: date/time: 12-Mar-2007/12:25 PartHomework Problems:-1 math 280B Homework Homework 1. Due Monday, January 22, 2007 Hand in from p. 114 : Hand in from p. 196 : , Hand in from p. 234 246: , , , (assume eachXnis inte-grable!), and For , observe thatXnd= nN(0,1).2. For , let{Un:n= 0,1,2,..}be random variables uniformlydistributed on (0,1) and takeX0=U0and then defineXninductively sothatXn+1=Xn Un+ For ; use the assumptions to boundE[Xn] in terms ofE[Xn:Xn x].Then use the two series Homework 2. Due Monday, January 29, 2007 Resnick Chapter 7:Hand , Resnick Chapter 7:look (For 28b, assumeE[XiXj] (i j) fori you may find it easier to showSnn 0 inL2rather than theweaker notion of in probability .)
8 Hand inExercise from these Notes . Resnick Chapter 8:Hand , (Assume Var (Nn)>0 for alln.) Homework #3 Due Monday, February 5, 2007 Resnick Chapter 8:Look , , Resnick Chapter 8:Hand , , , * (Due first), *Ignore the part of the question referring to the moment generating :use problem and the convergence of types theorem. Also hand in Exercise from these Homework #4 Due Friday, February 16, 2007 Resnick Chapter 9:Look , Resnick Chapter 9:Hand , , a-e., Also hand in Exercise from these Notes : , , and Homework #5 Due Friday, February 23, 2007 Resnick Chapter 9:Look at:8 Resnick Chapter 9:Hand in11, 28, 34 (assume n 2n>0), 35 (hint: showP[ n6= 0 ] = 0.), 38 (Hint: make use Proposition ) Homework #6 Due Monday, March 5, 2007 Look at Resnick Chapter 10: 11 Hand inthe following Exercises from the Lecture Notes : , , , , Resnick Chapter 10:Hand in2 , 5*, 7, 8** In part 2b, please explain what convention you are using when the denom-inator is 0.
9 *A Poisson process,{N(t)}t 0,with parameter satisfies (by definition): (i)Nhasindependent increments, so thatN(s) andN(t) N(s) are independent;(ii) if 0 u < vthenN(v) N(u) has the Poisson distribution with parameter (v u).**Hint:use Exercise Homework #7 Due Monday, March 12, 2007 Hand inthe following Exercises from the Lecture Notes : , , , Hand inResnick Chapter 10: 14 (takeBn:= (Y0,Y1,..,Yn) for thefiltration), Homework #8 Due Wednesday, March 21, 2007 by11:00AM! Look atthe following Exercise from the Lecture Notes : Hand inthe following Exercises from the Lecture Notes : Resnick Chapter 10:Hand in15, 28, and #28, letBn:= (Y1,..,Yn) define the :for part bconsider, 280A Homework ProblemsUnless otherwise noted, all problems are from Resnick, S. A ProbabilityPath, Birkhauser, Homework 1. Due Friday, September 29, 2006 p. 20-27: Look at: 9, 12, ,19, 27, 30, 36 p.
10 20-27: Hand in: 5, 17, 18, 23, 40, Homework 2. Due Friday, October 6, 2006 p. 63-70: Look at: 18 p. 63-70: Hand in: 3, 6, 7, 11, 13 and the following ( ).Referring to the setup in Problem 7 on p. 64 ofResnick, compute the expected number of different coupons collected after buy-ingnboxes of Homework 3. Due Friday, October 13, 2006 Look at from p. 63-70: 5, 14, 19 Look at Lecture Notes : exercise and read Section Hand in from p. 63-70: 16 Hand in Lecture note exercises: , and Homework 4. Due Friday, October 20, 2006 Look at from p. 85 90: 3, 7, 12, 17, 21 Hand in from p. 85 90: 4, 6, 8, 9, 15 Also hand in the following ( ).Suppose{fn} n=1is a sequence of Random Vari-ables on some measurable space. LetBbe the set of such thatfn( ) isconvergent asn .Show the setBis measurable, in the Homework 5. Due Friday, October 27, 2006 Look at from p.
