Transcription of Math 280 (Probability Theory) Lecture Notes
1 Bruce K. DriverMath 280 ( probability theory ) Lecture NotesApril 4, 2007 Homework Problems:-2 math 280C Homework Problems.. Homework #1 (Due Friday, April 14, 2007) .. 3-1 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 Wednesday, March 21, 2007 by 11:00AM! . 60 math 280A Homework Problems.. Homework 1. Due Friday, September 29, 2006 .. Homework 2. Due Friday, October 6, 2006.
2 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 .. Corrections and comments on Homework 7 (280A) .. Homework 8. Due Monday, November 27, 2006 .. Homework 9. Due Noon, on Wednesday, December 6, 2006 .. 9 Part I Background Material1 Limsups, Liminfs and Extended Limits..134 Contents2 Basic Probabilistic Notions..19 Part II Formal Development3 Preliminaries.. Set Operations .. Exercises .. Algebraic sub-structures of sets .. 314 Finitely Additive Measures.. Finitely Additive Measures.
3 Examples of Measures.. Simple Integration .. Simple Independence and the Weak Law of Large Numbers .. Constructing Finitely Additive Measures .. 525 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 .. A Baby Version of Kolmogorov s Extension Theorem.. 706 Random Variables.. Measurable Functions .. Factoring Random Variables .. 847 Independence.. and Monotone Class Theorems .. The Monotone Class Theorem .. Basic Properties of Independence .. An Example of Ranks .. Borel-Cantelli Lemmas.
4 Kolmogorov and Hewitt-Savage Zero-One Laws.. 1118 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 .. Comparison of the Lebesgue and the Riemann Integral .. Exercises .. Laws of Large Numbers Exercises .. 149 Contents59 Functional Forms of the Theorem..15110 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 The Polar Decomposition of Lebesgue Measure.
5 More Spherical Coordinates .. Exercises .. 19011Lp 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 .. 216 Part III Convergence Results12 Laws of Large Numbers.. Main Results .. Examples.. Random Series Examples .. A WLLN Example .. Strong Law of Large Number Examples .. 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.
6 25313 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 .. 2846 Contents14 Characteristic Functions (Fourier Transform).. Basic Properties of the Characteristic Function .. Examples.. Continuity Theorem .. A Fourier Transform Inversion Formula .. Exercises .. Appendix: Bochner s Theorem .. Appendix: A Multi-dimensional Weirstrass ApproximationTheorem .. Appendix: Some Calculus Estimates .. 32115 Weak Convergence of Random Sums.. Infinitely Divisible and Stable Symmetric Distributions.
7 Stable Laws .. 340 Part IV Conditional Expectations and Martingales16 Hilbert Space Basics.. Compactness Results forLp Spaces .. Exercises .. 35517 The Radon-Nikodym Theorem..35718 Conditional Expectation.. Examples.. Additional Properties of Conditional Expectations .. Regular Conditional Distributions .. Appendix: Standard Borel Spaces.. 37919 (Sub and Super) Martingales.. (Sub and Super) Martingale Examples .. Decompositions .. Stopping Times .. Stochastic Integrals and Optional Stopping .. Submartingale Inequalities .. Maximal Inequalities.. Upcrossing Inequalities and Convergence Results .. Supermartingale inequalities .. Maximal Inequalities.. The upcrossing inequality and convergence result.
8 Martingale Closure and Regularity Results.. Backwards Submartingales .. Appendix: Some Alternate Proofs .. 424 PartHomework Problems:-2 math 280C Homework Homework #1 (Due Friday, April 14, 2007) Look atthe following Exercises from the Lecture Notes : , , Look atthe following Exercises from Resnick Chapter 10: 19, 22-24, Hand inthe following Exercises from the Lecture Notes : , , , , 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].
9 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 .) 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 math 280B Homework Homework #4 Due Friday, February 16, 2007 Resnick Chapter 9:Look , Resnick Chapter 9:Hand , , a-e.
10 , 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 de-nominator is 0.*A Poisson process,{N(t)}t 0,with parameter satisfies (by definition):(i)Nhasindependent increments, so thatN(s) andN(t) N(s) are inde-pendent; (ii) if 0 u < vthenN(v) N(u) has the Poisson distribution withparameter (v u).
