1

Prove that if $\displaystyle X_n \rightarrow X$ in probability than $\displaystyle X_n \Rightarrow X$ (in distribution).

My approach:

$\displaystyle F_{X_n}(t) = P\{X_n \le t \} = $

$\displaystyle = P\{X_n \le t,\quad X \le t+\epsilon \} + P\{X_n \le t, \quad X > t+\epsilon \} $

$\displaystyle \le P\{X \le t+\epsilon \} + P\{X_n \le t, \quad X > t+\epsilon \}$

Because $\displaystyle X_n \rightarrow X$ in probability $\displaystyle \lim_{n \rightarrow \infty} P\{X_n \le t, \quad X > t+\epsilon \} = 0 $

Thus:

$\displaystyle \limsup_{n \rightarrow \infty}F_{X_n}(t) \le F_{X}(t+\epsilon)$

Hence: $\displaystyle \limsup_{n \rightarrow \infty}F_{X_n}(t) \le F_{X}(t)$

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 $\displaystyle Y_n$ be random variables with characteristic functions $\displaystyle \phi_n$, $\displaystyle n = 1, 2,...$

Prove that $\displaystyle Y_n \Rightarrow 0$ (in distribution) if there exists $\displaystyle \delta > 0$ such that $\displaystyle \lim_{n\rightarrow\infty} \phi_n(t) = 1$ for $\displaystyle |t| < \delta$.

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?