Random Process Lecture 1. Fundamentals of Probability

Transcription

1 Random Process Lecture 1. Fundamentals of Probability Husheng Li Min Kao Department of Electrical Engineering and Computer Science University of Tennessee, Knoxville Spring, /43

2 Outline 2/43 1 Syllabus 2 Measure Theory 3 Probability Spaces 4 Convergence 5 Conditioning

3 Syllabus Homework: 4 problems each week; hand in your homework after one week. 40% Midterm and Final Exames: 60% 3/43

4 Goal of This Course Understand the underlying math for probability and random processes measure theory Understand the basic concepts of probabilities such as distribution, expectation and convergence. Understand various types of random processes Know the applications in engineering 4/43

5 Outline Lecture 1. Measure Theory and Probability Lecture 2. Martingales Lecture 3. Markov Processes Lecture 4. Poisson Processes Lecture 5. Brownian Motion Lecture 6. Levy Processes Lecture 7. Stationary Process and Power Spectrum 5/43

6 Textbook E. Cinlar, Probability and Stochastics, Springer, /43

7 Outline 7/43 1 Syllabus 2 Measure Theory 3 Probability Spaces 4 Convergence 5 Conditioning

8 Sigma Algebra A nonempty collection E of subsets of set E is called an algebra provided that it is closed under finite unions and complements. It is called a σ-algebra if it is closed under complements and countable unions: A E E/A E. and A1, A2,... E n An E. Why σ-algebra? 8/43

9 9/43 Measurable Spaces and Functions A measurable space is a pair (E, E) where E is a set and E is a σ-algebra on E. Product of measurable spaces (E, E) and (F, F): (E F, E F). A mapping f : E F is said to be measurable relative to E and F if f 1 (B) E for any B F.

10 Indicators and Simple Functions Let A E, its indicator 1A is defined as 1, if x A, 1A (x) =. 0, otherise. A function f is called simple if f = n X i=1 10/43 ai 1Ai

11 Measures A measure µ on (E, E) is a mapping from E to nonnegative real numbers such that µ(φ) = 0, and µ( n An ) = X µ(an ), n for every disjoint sequence {An }. A measure space is a triplet (E, E, µ). Almost everywhere: If a proposition holds for all but a negligible set of x in E, then we say that it holds for almost every x, or almost everywhere. 11/43

12 Example of Measures Dirac measures. Counting measures. Discrete measures. Lebesgue measures. 12/43

13 Integration When f is simple and positive. Then we define its integration as µf = n X ai µ(ai ). i=1 For positive measurable function f, we can use a series of simple functions to approach it fn f Then we define µf = lim µfn. For generic measurable function f, we define µf = µ(f + ) µ(f ). If µf exists and is a real number, we say that f is measurable. 13/43

14 Examples of Integrations Discrete measures. Discrete spaces. Lebesgue integrals. Integral of Dirichlet function. 14/43

15 Intuition of Lebesgue Integrals: Counting Moneys Lebesgue integral and Riemann integral imply different approaches of counting money. 15/43

16 Convergence Theorems Monotone Convergence Theorem: Let fn be increasing in E+, then µ(lim fn ) = lim µfn. Dominated Convergence Theorem: Suppose {fn } is dominated by some integrable function g. If lim fn exists, then it is integrable and µ(lim fn ) = lim µfn. Fatou s Lemma: if {fn } E+, then µ(lim inf fn ) lim inf µfn. 16/43

17 Differentiation Let µ and ν be measures on a measurable space. Then, ν is said to be absolutely continuous with respect to µ, if for every set A E µ(a) = 0 ν(a) = 0. Radon-Nikodym Theorem: Suppose µ is σ-finite, and ν is absolutely continuous w.r.t. µ. THen there exists a positive measurable function p such that Z Z ν(dx)f (x) = µ(dx)p(x)f (x). We can denote p by dν/dµ and call it Radon-Nikodym derivative. 17/43

18 Kernel For measurable spaces (E, E) and (F, F), K, a mapping from E F to R+, is called a kernel if the mapping x K (x, B) is E-measurable for every set B in F. the mapping B K (x, B) is a measure on (F, F) for every x in E. 18/43

19 Outline 19/43 1 Syllabus 2 Measure Theory 3 Probability Spaces 4 Convergence 5 Conditioning

20 Probability Space Let (Ω, H, P) be a probability. Ω is called the sample space, its elements are called the outcomes. H is called the event space. An event is said to be almost sure if its probability is 1. 20/43

21 Random Variables Let (E, E) be a measurable space. A mapping X : Ω E is called a random variable if X 1 A is measurable in H. Distribution: µ(a) = P(X 1 A), for A E. Independence: P(X A, Y B) = P(X A)P(Y B). 21/43

22 Stochastic Processes Let (E), E be a measurable space. Let T be an arbitrary set, countable or uncountable. For any t in T, let Xt be a random variable taking values in (E, E). Then, the collection {Xt, t T } is called a stochastic process with state space (E, E). One can consider the random process Xt as a random variable taking values in the product space (E T, E T ). The distribution of random process is determined by probability of finite samples: P(Xt1 A1,..., Xtn An ). 22/43

23 Expectations The expectation of a random variable X is given by Z EX = P(dw)X (w). Ω If X takes values in a measurable space (E, E) and µ is the distribution of X, then we have E[f (X )] = µf. 23/43

24 Lp -Space Define the p-norm of random variable X : kx kp = (E X p )1/p. Let Lp denote the collection of all real-valued random variables X with kx kp <. Holder s inequality: if 1/p + 1/q = 1/r, then kxy kr kx kp ky kq. Mikowski s Inequality: for p 1 kx + Y kp kx kp + ky kp. 24/43

25 Uniform Integrability A collection K of real valued random variables is said to be uniformly integrable if lim = sup E[ X 1( X > b)]. b X K Necessary and sufficient condition for uniform integrability: for any > 0 we can find a δ > 0 such that for every event H P(H) δ sup E X 1H X K 25/43

26 Information and Determinability We call the set {X 1 A, A E} the σ-algebra generated by X. For a collection of random variables {Xt t T }, we define the σ-algebra generated by {Xt } as σ{xt : t T } = t T σxt. A mapping V belongs to σx if and only if V = f (X ) for some deterministic function f in E. We can equate the information available from the realization of {Xt } to the σ-algebra generated by {Xt }. 26/43

27 Filtrations Let T be a subset of R. For each t T, let Ft be a sub-σ-algebra of H. The family F = {Ft : t T } is called a filtration given that Fs Ft for s < t. A typical filtration: Ft = σ{xs : s t, t T }. 27/43

28 Independence Let F1,..., Fn be sub-σ-algebra of H. Then, {F1,..., Fn } is called an independency if E[V1...Vn ] = E[V1 ]...E[Vn ], for all positive random variables V1,..., Vn in F1,..., Fn respectively. F1,..., Fn,... of H are independent if and only if F1,...,n and Fn+1 are independent. Independent random variables: {Xt, t T } is independent if σ{xt, t T } is an independency. 28/43

29 0-1 Laws Let Gn be a sequence of sub-σ-algebra of H. Define Fn = m>n Gm and the tail-σ-algebra F = n Fn. Kolmogorov 0-1 Law: Let G1,..., Gn,... be independent. Then P(H) is either 0 or 1 for every event H in the tail F. Consider a permutation p and X = (X1, X2,...). Then, X p = (Xp(1),..., Xp(2),...). A random variable V in F is said to be permutation invariant if V p = V. Hewit 0-1 Law: Suppose that X1,..., Xn,... are i.i.d. Then every permutation invariant event has probability 0 or 1. 29/43

30 Outline 30/43 1 Syllabus 2 Measure Theory 3 Probability Spaces 4 Convergence 5 Conditioning

31 Homework Problems Deadline: Feb. 2, 2016 Problem 1: Let F be a σ-algebra of subsets of Ω. Suppose B F. Show that G = {A B : A F} is a σ-algebra of subsets of B. Problem 2: Consider a random variable mapping from (Ω, H, P) to (E, E). Prove that the set generated by X, {X 1 (A) A E} is a σ-algebra. Problem 3: Use both the Riemann integral and Lebesgue integral to R1 evaluate 0 1dx. Problem 4: Show that Kolmogorov 0-1 Law implies the law of large numbers. 31/43

32 Almost Sure Convergence A sequence {Xn } is said to be almost surely convergent if the numerical sequence {Xn (w)} is convergent for almost every w. Borel-Cantelli s Lemma: Let Hn be a sequence of events. Then X X P(Hn ) < 1Hn < almostsurely. If 32/43 P n P( Xn X > ) <, then Xn X almost surely.

33 Convergence in Probability The sequence of random variables {Xn } converges to X in probability if for every > 0 we have lim P Xn X > = 0. n If Xn converges almost surely, then it converges in probability. If Xn converges in probability, then it has a subsequence converging almost surely. 33/43

34 Convergence in Lp The sequence of random variables {Xn } converges to X in Lp if every Xn and X are in Lp and lim E Xn X p = 0. n The following are equivalent: {Xn } converges in L1 It converges in probability and is uniformly integrable. It is Cauchy for convergence in L1. 34/43

35 Weak Convergence Let {Xn } has distributions {µn }. {µn } is said to converge weakly to µ if lim µn f = µf, for any f in Cb. The sequence {Xn } is said to converge to X if µn converges to µ. A sequence µn is said to be tight if for every > 0 there is a compact set K such that µn (K ) > 1 for all n. Theorem: If {µn } is tight then every subsequence of it has a further subsequence that is weakly convergent. 35/43

36 Law of Large Numbers For {Xi }, define Sn = X Xn and X n = n1 Sn. Theorem: If {Xn } is pairwise i.i.d, then X n convergences in L2, in probability and almost surely. Theorem: If {Xi } is pairwise independent and have the same distribution as a generic variable X. Then, if EX exists, X converges to EX almost surely. 36/43

37 Central Limit Theorem Let {Xi } be i.i.d. Let Sn = Pn i=1 Xi and Zn = S n ne[xi ]. Then, Zn nvar [Xi ] converge to a standard Gaussian random variable in distribution. The law of large numbers and the central limit theorem are the most important theorems in probably theory. There are many variants of these two fundamental theorems (e.g., what if the samples are not i.i.d?). 37/43

38 Outline 38/43 1 Syllabus 2 Measure Theory 3 Probability Spaces 4 Convergence 5 Conditioning

39 Conditional Expectations Let F be a sub-σ-algebra of H. The conditional expectation of X given F denoted by EF X is defined in two steps: For X in H+, the conditional expectation is defined as any random variable X that satisfies (1) X belongs to F+ and EVX = EVX for every V in F+. FOr arbitrary X in H, we define EF (X ) = EF (X + ) EF (X ). 39/43

40 40/43 Special Properties Let F and G be sub-σ-algebra of H. Let W and X be random variables (in H) such that EX and EWX exist. Then we have Conditional determinism: W G EF WX = WEF X. Repeated conditioning: F EF EG X = EG EF X = EF X

41 Conditional Expectations Given Random Variable Recall σy is the σ-algebra generated by random variable Y. Then, we define E[X Y ] = EσY X. 41/43

42 Conditional Distributions Let F be a sub-σ-algebra of H. For each H in H, PF H = EF 1H is called the conditional probability of H given F. Let Q(H) be a version of PF H for H in H. Then, Q is said to be a regular version of the conditional probability if Q is a transition probability kernel from (Ω, F) into (Ω, H). 42/43

43 Conditional Independence Let {Fi } be sub-σ-algebras of H. Then they are said to be conditional independent given F if EF V1...Vn = EF V1...EF Vn. 43/43