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1、Math 323Fall 2001 Richard Bass 1. Basics of probability. In this section we give some preliminaries on probabilistic terminology, indepen- dence, and Gaussian random variables. Given a space and a -fi eld (or -algebra) F on it, a probability measure, or just a probability, is a positive fi nite meas
2、ure P with total mass 1. (,F,P) is called a probability space. Elements of F are called events. Measurable functions from to R are called random variables and are usually denoted X or Y instead of f or g. The integral of X with respect to P is called the expectation of X or the expected value of X,
3、and R X()P(d) is often written EX, while R A X()P(d) is often written EX;A. If an event occurs with probability one, one says “almost surely” and writes a.s. The indicator of the set A is the function or random variable denoted 1Athat is 1 on A and 0 on the complement. The complement of A is denoted
4、 Ac. If Anis a sequence of sets, (An i.o.), read “infi nitely often,” is defi ned to be j=1n=jAn. The following easy fact is called the Borel-Cantelli lemma or sometimes the fi rst half of the Borel-Cantelli lemma. Proposition 1.1. If P n=1P(An) )d = E Z X 0 pp1d, which is equal to the left-hand sid
5、e. Two events A, B are independent if P(A B) = P(A)P(B). This defi nition gener- alizes to n events: A1,.,Anare independent if P(Ai1 Ai2 Aij) = P(Ai1)P(Ai2)P(Aij) for every subset i1,.,ij of 1,2,.,n. A -fi eld F is independent of a -fi eld G if each A F is independent of each B G. The -fi eld genera
6、ted by X, denoted (X), is the collection (X A);A Borel. Two random variables are independent if the - fi elds generated by X, Y are independent. The notion of an event and a random variable being independent, or a random variable and a -fi eld being independent are defi ned in the obvious way. Note
7、that if X and Y are independent and f and g are Borel measurable functions, then f(X) and g(Y ) are independent. An example of independent random variables is to let = 0,12, P Lebesgue measure, X a function of just the fi rst variable, and Y a function of just the second variable. In fact, it can be
8、 shown that independent random variables can always be represented by means of some suitable product space. Proposition 1.6. If X, Y , and XY are integrable and X and Y are independent, then EXY = (EX)(EY ). Proof.If X is of the form PI i=1ai1Ai, Y is of the form PJ j=1bj1Bj and X and Y are independ
9、ent, then by linearity and the defi nition of independence, EXY = (EX)(EY ). Approximating nonnegative X and Y by simple random variables of this form, we obtain our result by monotone convergence. The case of general X and Y then follows by linearity. The characteristic function of a random variabl
10、e X is the Fourier transform of its law: R eiuxPX(dx) = EeiuX(we are using Proposition 1.4 here). If X and Y are inde- pendent, so are eiuXand eivY, and hence by Proposition 1.6, Eei(uX+vY )= EeiuXEeivY. Thus when X and Y are independent, the joint characteristic function of X and Y factors into the
11、 product of the respective characteristic functions. The converse also holds. 3 Proposition 1.7. If Eei(uX+vY )= EeiuXEeivYfor all u and v, then X and Y are independent random variables. Proof.Let X0be a random variable with the same law as X, Y 0 one with the same law as Y , and X0, Y 0 independent
12、.(We let = 0,12, P Lebesgue measure, X0a function of the fi rst variable, and Y 0 a function of the second variable defi ned as in (1.2).) Then Eei(uX 0+vY0) = EeiuX 0EeivY0. Since X,X0 have the same law, they have the same characteristic function, and similarly for Y,Y 0. Therefore (X0,Y 0) has the
13、 same joint characteristic function as (X,Y ). By the uniqueness of the Fourier transform, (X0,Y 0) has the same joint law as (X,Y ), which is easily seen to imply that X and Y are independent. The second half of the Borel-Cantelli lemma is the following assertion. Proposition 1.8. If Anis a sequenc
14、e of independent events and P n=1P(An) = , then P(Ani.o.) = 1. Proof.Note P(N n=jAn) = 1 P( N n=jA c n) = 1 N Y n=j P(Ac n) = 1 N Y n=j (1 P(An) 1 exp ? N X n=j P(An) ? 1 as N . So P(Ani.o.) = limj( n=jAn) = 1. A mean zero, variance one, Gaussian or normal random variable is one where P(X A) = Z A 1
15、 2ex 2/2dx, A Borel.(1.5) We also describe such an X as having an N(0,1) distribution or law, read as a “normal 0,1” law. It is routine to check that such an X has mean or expectation 0 and variance E(X EX)2= 1. It is standard that EeiuX= eu 2/2. Such random variables exist by the construction in (1
16、.2), with (dx) = (2)1/2ex 2/2dx. X has an N(a,b2) distribution if X = bZ + a for some Z having a N(0,1) distribution. A sequence of random variables X1,.,Xnis said to be jointly normal if there exists a sequence of independent N(0,1) random variables Z1,.,Zmand constants bij 4 and aisuch that Xi= Pm j=1bijZj+ai, i = 1,.,n. In matrix notation, X = BZ+A. For simplicity, in what follows let us ta
