Convergence of Random Variables Probability Inequalities

Transcription

1 Convergence, MIT Dr. Kempthorne Spring MIT

2 Outline Convergence, 1 Convergence, 2 MIT

3 Convergence, /Vectors Framework Z n = (Z n,1, Z n,2,..., Z n,d ) T, a sequence of random vectors. Z = (Z 1, Z 2,..., Z d ) T, a random vector (e.g., sequence limit) Definitions/Terminology/Theorems (B.7.1) Convergence in robability : {Z n } converges in probability to Z. Z n Z 0 Z n Z Definition: For every E > 0: lim ( Z n Z > E) 0. n NOTE(!): Convergence in robability REQUIRES joint distribution of Z n and Z. 3 MIT

4 Convergence, /Vectors Definitions/Terminology (continued) (B.7.2) Convergence in aw / Convergence in Distribution: {Z n } converges in law to Z. Z n Z (Z n ) (Z) Definition: for every t R d, where the distribution function F Z of Z is continuous: lim F Zn (t) = F Z (t). n NOTE(!): Convergence in aw/distribution does NOT use joint distribution of Z n and Z. (A.14.4) If Z = z 0, a constant, convergence in law/distribution implies convergence in probability: Z n z 0 = Z n z 0. 4 MIT

5 Convergence, (A.14.6) If Z n z 0, and g is continuous at z 0, then g(z n ) g(z 0 ). (A.14.8) If Z n Z, and g is continuous, then g(z n ) g(z ). Theorem (A.14.9) If Z n Z, and U n u0, a constant, then (a). Z n + U n Z + u 0, (b). U n Z n u 0 Z. 5 MIT

6 Convergence, Corollary (A.14.17) Suppose {a n }: lim n a n =. b: < b <, a fixed number. a n (Z n b) Z. g( ) : a function of a real variable whose derivative, g ', exists and is continuous at b. Then a n [g(z n ) g(b)] g ' (b)z. 6 MIT

7 Convergence, /Vectors Theorem B.7.2 Slutsky s Theorem. Suppose Z T T n = (U n, V n ) where Z n is a d-vector, U n is a b-vector, V n is a c-vector (d = b + c) U n U v, a constant vector V n g : R d R b is continuous Then T T g(u, V g(u T n ), V T n ). Examples: (a). d = 2, b = c = 1, g(u, v) = αu + βv, or g(u, v) = u/v T 7 MIT

8 Slutsky s Theorem Convergence, Examples (continued): (b). V n U n + W n vu + w, where {V n } matrix r.v. s (r d) V n v (constant matrix) U n U W n w (constant vector) (all dimensions conformal) Theorem B.7.4 If Then: U n U g( ): bounded and [U A g ] = 1, Eg(U n ) Eg(U). 8 MIT

9 Convergence, Dominated Convergence Theorem Theorem B.7.5 Dominated Convergence Theorem. If {W n }, W and V are random variables with W n W ( W n < V ) = 1, for all n E [ V ] < Then: E [W n ] E [W ]. 9 MIT

10 Outline Convergence, 1 Convergence, 10 MIT

11 Inequalities Convergence, (a). Chebychev s Inequality: If X is any random variable, then E [X 2 ] [ X a]. a 2 (b). Markov s Inequality: If X is any random variable, then E [ X ] [ X a]. a (c). Generalization: If X is any random variable, and g( ) is non-negative and non-decreasing on range of X : E[g(X )] [X a]. g(a) (d). Bernstein s Inequality: g(t) = e st : If X is any random variable, E[e sx ] [X a]. e sa 11 MIT

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