文档介绍：Outline
Characteristic Function and Its Properties
Distribution Determined by Characteristic Function
Chapter 4: Characteristic Function
Jiangsheng Yu
School of Electronics Engineering puter Science
Peking University, Beijing 100871, China
Probability Theory and Mathematical Statistics, 2007-2008
Jiangsheng Yu Characteristic Function
Outline
Characteristic Function and Its Properties
Distribution Determined by Characteristic Function
Outline of topics
1 Characteristic Function and Its Properties
Definition of characteristic function
Properties of characteristic function
Relationship between characteristic function and moments
Semi-invariants
Characteristic function of sum of independent variables
Characteristic function can not declare independence
2 Distribution Determined by Characteristic Function
Levy´ theorem
Describe density by characteristic function
Gnedenko’s counterexample
Jiangsheng Yu Characteristic Function
Outline Definition of characteristic function
Characteristic Function and Its Properties Properties of characteristic function
Distribution Determined by Characteristic Function Characteristic function can not declare independence
Definition of characteristic function for random variable
Definition (Characteristic Function of A Random Variable)
Let X be a random variable, the function
ϕ(t) = E(eitX ) (1)
is called the characteristic function of X, where t ∈ R.
For example, P(X = −1) = P(X = 1) = , we have
ϕ(t) = −it +
= cos t
Jiangsheng Yu Characteristic Function
Outline Definition of characteristic function
Characteristic Function and Its Properties Properties of characteristic function
Distribution Determined by Characteristic Function Characteristic function can not declare independence
Definition of characteristic function for random vector
Definition (Characteristic Function of A Random Vector)
Let (X, Y) be a random vector, the function
ϕ(s, t) = E(ei(sX+tY))