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1 Advaced Probability Theory Math5411 HKUST Kai Che (Istructor)

2 Chapter 1. Law of Large Numbers 1.1. σ-algebra, measure, probability space ad radom variables. This sectio lays the ecessary rigorous foudatio for probability as a mathematical theory. It begis with sets, relatios amog sets, measuremet of sets ad fuctios defied o the sets. Example 1.1. (A prototype of probability space.) Drop a eedle blidly o the iterval [0, 1]. The eedle hits iterval [a, b], a sub-iterval of [0, 1] with chace b a. Suppose A is ay subset of [0, 1]. What s the chace or legth of A? Here, we might iterprete the largest set Ω = [0, 1] as the uiverse. Note that ot all subsets are ice i the sese that their volume/legth ca be properly assiged. So we first focus our attetio o certai class of ice subsets. To begi with, the Basic subsets are all the sub-itervals of [0, 1], which may be deoted as [a, b], with 0 a b 1. Deote B as the collectio of all subsets of [0, 1], which are geerated by all basic sets after fiite set operatios. B is called a algebra of Ω. It ca be proved that ay set i B is a fiite uio of disjoit itervals (closed, ope or half-closed). Still, B is ot rich eough. For example, it does ot cotai the set of all ratioal umbers. More importatly, the limits of sets i B are ofte ot i B. This is serious restrictios of mathematical aalysis. Let A be the collectio of all subsets of [0, 1], which are geerated by all basic sets after coutably may set operatios. A is called Borel σ-algebra of Ω. Sets i A are called Borel sets. Limits of sets i A are still i A. (Ω, A) is a measurable space. Borel measure: ay set A i A ca be assiged a volume, deoted as µ(a), such that (i). µ([a, b]) = b a. (ii). µ(a) = lim µ(a ) for ay sequece of Borel sets A A. Lebesgue measure (1901): Completio of Borel σ-algebra by addig all subsets of Borel measure 0 sets, deoted as F. Sets with measure 0 are called ull sets. Why should Borel measure or Lebesgue measure exist i geeral? Caratheodory s extesio theorem: extedig a (σ-fiite) measure o a algebra B to the σ-algebra A = σ(b). Ω = [0, 1] (the uiverse). B: a algebra (fiite set operatios) geerated by subitervals. A: the Borel σ-algebra, is a σ-algebra, geerated by subitervals. F: completio of A, a σ-algebra, geerated by A ad ull sets. (Ω, B, µ) does ot form a probability space, (Ω, A, µ) forms a probability space. (Ω, F, µ) forms a probability space. Sets ad set operatios: Cosider Ω as the uiverse, (Beyod which is othig.) Write Ω = {ω}, ω deotes a member of the set, called elemet. Let A ad B: be two subsets of Ω, called evets. The set operatios are: itersectio:, A B: both A ad B (happes). uio:, A B: either A or B (happes).

3 complemet: A c = Ω \ A: everythig except for A, or A does ot happe. mius: A \ B = A B c : A but ot B. A elemetary theorem about set operatio is DeMorga s idetity: ( j=1a j ) c = j=1 A c j, ( j=1a j ) c = j=1 A c j. I particular, (A B) c = (A c B c ), i.e., (A B) c = (A c B c ). Remark. Itersectio ca be geerated by complemet ad uio; ad uio ca be geerated by complemet ad itersectio. Relatio: A B, if ω A esures ω B. A sequece of sets {A : 1} is called icreasig (decreasig) if A A +1 (A A +1.) A = B if ad oly if A B ad B A. Idicator fuctios. (A very useful tool to traslate set operatio ito umerical operatio) The relatio ad operatio of sets are equivalet to the idicatio set fuctios. For ay subset A Ω, defie its idicator fuctio as { 1 if ω A 1 A (ω) = 0 Otherwise. The idicator fuctio is a fuctio defied o Ω. Set operatios vs. fuctio operatios: A B 1 A 1 B. A B 1 A 1 B = 1 A B = mi(1 A, 1 B ). A c = Ω \ A 1 1 A = 1 A c. A B 1 A B = 1 A + 1 B, if A B = 1 A B = max(1 A, 1 B ). Set limits. There are two limits of sets: upper limit ad low limit. lim sup A =1 k= A k = {A ifiitely occurs.} 1 lim sup A = lim sup 1 A ω lim sup A if ad oly if ω belogs to ifiitely may A. Lower limit. lim if A =1 k= A k = {A always occurs except for fiite umber of times.} 1 lim if A = lim if 1 A ω lim if A if ad oly if ω belogs to all but fiitely may A. We say the set limit of A 1, A 2,... exists if their lower limit is the same as the upper limit. Algebra ad σ-algebra

4 A is a o-empty collectio (set) of subsets of Ω. Defiitio. A is called a algebra if (i). A c A if A A; (ii). A B A if A, B A. A is called a σ-algebra if, (ii) is stregtheed as, (iii). =1A A if A A for 1. A algebra is closed for (fiite) set operatios. Ω A ad A. A σ-algebra is closed for coutable operatios. (Ω, A) is called a measurable space, if A is a σ-algebra of Ω. Measure, measure space ad probability space. A, cotaiig, is a o-empty collectio (set) of subsets of Ω. µ is a oegative set fuctio o A. µ is called a measure, if (i). µ( ) = 0. (ii). µ(a) = =1 µ(a ) if A, A 1, A 2,... are all i A ad A 1, A 2,... are disjoit. (Ω, A, µ) is called a measure space, if µ is a measure o A ad A is a σ-algebra of Ω. (Ω, A, P ) is called a probability space if (Ω, A, P ) is a measure space ad P (Ω) = 1. For probability space (Ω, A, P ), Ω is called sample space, every A i A is a evet, ad P (A) is the probability of the evet, the chace that it happes. Radom variable (r.v.). Loosely speakig, give a probability space (Ω, F, P ), a radom variable (r.v.) X is defied as a real-valued fuctio of Ω, satisfyig certai measurability coditio. Loosely speakig, viewig X = X(ω) as a mappig from Ω to R, the real lie, the X 1 (B) must be i F for all Borel sets B. (Borel sets o real lie are the σ-algebra geerated by itervals, i.e., the sets geerated by coutable operatios o itervals). A radom variable X defied o a probability space (Ω, A, P ) is a fuctio defied o Ω, such that X 1 (B) A for every iterval B o [, ], where X 1 (B) = {ω : X(ω) B}. (We eed to idetify its probability.) X 1 (B) is called the iverse image of B. X = X( ) ca be viewed as a map or trasformatio from (Ω, A) to (R, B), where R = [, ] ad B is the σ-algebra geerated by the itervals i R. X is a measurable map/trasformatio sice X 1 (B) A for every B B (DIY.) Because A is a σ-algebra, the upper ad lower limits of X is a r.v. algebraic operatios: +,,, /, of r.v.s are still r.v.s. if X are r.v.s., ad the Measurable map ad radom vectors. f( ) is called a measurable map/trasformatio/fuctio from a measurable space (Ω, A) to aother measurable space (S, S), if f 1 (B) A for every B S. i.e. {w : f(w) B} A. X is called a radom vector of p dimesio if it is a measurable map from a probability space (Ω, A, P ) to (R p, B p ), where B p is the Borel σ-algebra i p dimesioal real space, R p = [, ] p.

5 Propositio 1.1 ((2.3) i the textbook.) If X = (X 1,..., X p ) is a radom vector of p dimesio o a probability space (Ω, A, P ), ad f( ) is measurable fuctio from (R p, B p ) to (R, B), the f(x) is a radom variable. Proof. For ay Borel set B B, {ω : f(x(ω)) B} = {ω : X(ω) f 1 (B)} A sice f 1 (B) B p. Propositio 1.2 ((2.5) i the textbook.) If X 1, X 2,... are r.v.s. So are if X, sup X lim if X ad lim sup X. Proof. Let the probability space be (Ω, A, P ). For ay x, {ω : if X (ω) x} = {ω : X (ω) x} A; {ω : sup X (ω) x} = {ω : X (ω) x} A; {lim if X > x} = { if X k > x} A; k {lim sup X < x} = {sup X k < x} A. k Therefore, if X, sup X, lim if X ad lim sup X are r.v.s. Propositio 1.3 Suppose X is a map from a measurable space (Ω, A) to aother measurable space (S, S). If X 1 (C) A for every C C ad S = σ(c). The, X is a measurable map, i.e., X 1 (S) A for every S S. I particular, whe (S, S) = ([, ], B), X 1 ([, x]) A for every x is eough to esure X is a r.v.. Proof. Note that σ(c), the σ-algebra geerated by C, is defied mathematically as the smallest σ-algebra cotaiig C. Set B = {B S : X 1 (B) A}. We first show B is a σ-algebra. Observe that (i). for ay B B, X 1 (B) A ad, therefore, X 1 (B c ) = (X 1 (B)) c A; (ii). for ay B B, X 1 (B ) A ad X 1 ( B ) = X 1 (B ) A. Cosequetly, B is a σ-algebra. Sice C B S, it follows that B = S. Summary of Sectio 1.1 σ-algebra: collectio of sets which is closed uder coutably may set operatios. Probability space: The trio (Ω, A, P ) with A as a σ-algebra of Ω ad P a set fuctio such that (i) 0 P (A) 1 for ay A A ad P (Ω) = 1. (ii). P ( A ) = P (A ) for coutable disjoit A A. A radom variable X is a fuctio/map o Ω with value i [, ] such that {X [, x]} A. F (x) P (X x) is called (cumulative) distributio fuctio of X. The moral is to esure calibratio of the distributio of r.v.s ad validity of algebraic operatio ad limits of r.v.s. idicator fuctio as a useful tool. DIY Exercises: Exercise 1.1 Show 1 lim if A = lim if 1 A ad DeMorge s idetity.

6 5 Exercise 1.2 Show that, the so called coutable additivity or σ-additivity, (P ( A ) = P (A ) for coutable disjoit A A), is equivalet to fiite additivity plus cotiuity (if A, the P (A ) 0.) Exercise 1.3 (Completio of a Probability space) Let (Ω, F, P ) be a probability space. Defie F = {A : P (A \ B) + P (B \ A) = 0, for someb F}, Ad for each A F, P (A) is defied as P (B) for the B give above. Prove that (Ω, F, P ) is also a probability space. (Hit: eed to show that F is a σ-algebra ad that P is a probability measure.) Exercise 1.4 If X 1 ad X 2 are two r.v.s, so is X 1 + X 2. (Hit: cite Propositios 1.1 ad 1.3)

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