Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Prove the following theorem:

Let $\phi: \Bbb R \to \Bbb C$.

$\phi$ is the characteristic function of a real random variable $X:\Omega \to \Bbb R$ if and only if $\phi(0)=1$ $\phi$ is uniformly continuous at $0$ and it's semi positive definite.

I have no idea how Can I attack this problem, please give me some tips, and tools. Thanks

share|improve this question
    
The "only if" part follows trivially from Fourier theory. The Fourier transform of a Borel probability measure is positive definite, etc. The most clean argument for the "if" part is probably via Gelfand theory. –  Michael Dec 13 '13 at 9:19

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.