Statistics 330/600 2005: Sheet 1 Please attempt at least the starred problems. *(1.1) Please print clearly your name, year, and major or graduate department.
Due: Thursday 20 January
*(1.2) let R denote the set of all half-open rectangles of the form (a1
Contents
Preface
xi Why bother with measure theory? 1 The cost and benet of rigor 3 Where to start: probabilities or expectations? The de Finetti notation 7 Fair prices 11 Problems 13 Notes 14 Measures and sigma-elds 17 Measurable functions 22 Integrals 2
Strong Law of Large Numbers Theorem. (Etemadi 1981) Let X1 , X2 , . . . be independent, identically distributed, integrable random variables, with common expected value . Let Sn = n i=1 Xi . Then Sn /n as n almost surely. Proof. For each i N, dene bounded
Chapter 1
Motivation
SECTION 1 offers some reasons for why anyone who uses probability should know about the measure theoretic approach. SECTION 2 describes some of the added complications, and some of the compensating benets that come with the rigorous t
Chapter 2
A modicum of measure theory
2 February 2004: Modication of Section 2.11.
*1.
Generating classes of functions
Theorem <Dynkin.thm> is often used as the starting point for proving facts about measurable functions. One rst invokes the Theorem to es
Chapter 2
A modicum of measure theory
SECTION 1 denes measures and sigma-elds. SECTION 2 denes measurable functions. SECTION 3 denes the integral with respect to a measure as a linear functional on a cone of measurable functions. The denition sidesteps th
Preface
This book began life as a set of handwritten notes, distributed to students in my one-semester graduate course on probability theory, a course that had humble aims: to help the students understand results such as the strong law of large numbers, t
Chapter 4
Product spaces and independence
1.
<1>
Product measures
Denition. Let X1 , . . . , Xn be sets equipped with sigma-fields A1 , . . . , An . The set of all ordered n -tuples (x1 , . . . , xn ), with xi Xi for each i is denoted by X1 . . . Xn or Xi
Statistics 330/600 2005: Sheet 12 *(12.1) Let X be a metric space. (i) UGMTP Problem 7.7
Due: Thursday 21 April
(ii) Let P be a probability measure on B(X). Suppose cfw_ X n : n N is a sequence of random elements of X, with the following property: for eve
Statistics 330/600 2005: Sheet 11
Due: Thursday 14 April
*(11.1) Suppose cfw_ X n : n N0 is a sequence of N0 -valued random variables that converges almost surely to a random variable X . Suppose the sequence is adapted to a ltration cfw_Fn : n N0 and t
Statistics 330/600 2005: Sheet 10
Due: Thursday 7 April
*(10.1) Suppose X L1 ( , F, P) and Y = P( X | F0 ) for some sub-sigma-eld F0 of F. Suppose W is an F0 -measurable random variable for which X W L1 ( , F, P). (i) Show that Y W L1 ( , F, P) and P( X W
Statistics 330/600 2005: Sheet 2 Please attempt at least the starred problems.
Due: Thursday 27 January
*(2.1) (H lder inequality) UGMTP Problem 2.15 or 2.16, not both. Be careful with log 0. o *(2.2) (Minkowski inequality/Orlicz norm) UGMTP Problem 2.17
Statistics 330/600 2005: Sheet 3 Please attempt at least the starred problems. *(3.1) (separation of pairs of points) UGMTP Problem 2.5. *(3.2) Suppose X and Y are independent random variables on ( , F, P). That is, Pcfw_ X A , Y B = Pcfw_ X APcfw_Y B
D
Statistics 330/600 2005: Sheet 4
Due: Thursday 10 February
No lecture on Tuesday 8 February
*(4.1) (Weierstrass approximation) UGMTP Problem 2.25. Note: You would need this result to understand the handout on -spaces. Remember, a continuous function on [0
Statistics 330/600 2005: Sheet 5
Due: Thursday 17 February
*(5.1) Suppose Ei is a collection of subsets of Xi with Xi Ei , for i = 1, 2. Dene E1 E2 := cfw_ E 1 E 2 : E 1 E1 , E 2 E2 Show that E1 E2 = (E1 ) (E2 ), as sigma-elds on X1 X2 . Hint: Show that
Statistics 330/600 2005: Sheet 6
Due: Thursday 24 February
*(6.1) Find a probability measure on a set with only four points, and sets Ai such that P( A1 A2 A3 ) = (P A1 )(P A2 )(P A2 ) but for which the sigma-elds Gi := cfw_ Ai , for i = 1, 2, 3, are not
Statistics 330/600 2005: Sheet 7
Due: Thursday 3 March
*(7.1) (Integration by parts) UGMTP Problem 4.16, rst two parts. If you attempt the third part you should assume that has density f and has density g , both with respect to Lebesgue measure. You shoul
Statistics 330/600 2005: Sheet 8
Due: Thursday 24 March
*(8.1) Let and be nite measures on a sigma-eld of subsets of X, with . Let T be an A\B-measurable map into a set T equipped with a sigma-eld B. (i) Show that T T . Write g for the density d (T )/d (T
Statistics 330/600 2005: Sheet 9
Due: Thursday 31 March
*(9.1) (Pythagoras in L2 ) UGMTP problem 5.9. Note: the conditional variance, var( X | G), is dened as P ( X Y )2 | G where Y = P( X | G). (9.2) (measurable questionstrouble with the heuristic) UGMTP
Reference books Amongst the sources for my notes were the following books, all of which I recommend. Ash, R. B. Real Analysis and Probability Good background on measure theory, particularly the connections between topology and measure. Recommended for mar