1

Prove that if in probability than (in distribution).

My approach:

Because in probability

Thus:

Hence:

I was unable to prove the other inequality unless X is absolutly continuous.

Does anybody know how to do that?

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2.

Let be random variables with characteristic functions ,

Prove that (in distribution) if there exists such that for .

I was unable to solve it.The only reasonable idea I had was to assume that all moments of the variables exist and prove thay all converge to 0 (than convergense of distribution follows, I think)

Can you help me with this one too?