1. ## Some problems

1

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

My approach:

$F_{X_n}(t) = P\{X_n \le t \} =$
$= P\{X_n \le t,\quad X \le t+\epsilon \} + P\{X_n \le t, \quad X > t+\epsilon \}$
$\le P\{X \le t+\epsilon \} + P\{X_n \le t, \quad X > t+\epsilon \}$

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

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

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

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 $Y_n$ be random variables with characteristic functions $\phi_n$, $n = 1, 2,...$
Prove that $Y_n \Rightarrow 0$ (in distribution) if there exists $\delta > 0$ such that $\lim_{n\rightarrow\infty} \phi_n(t) = 1$ for $|t| < \delta$.

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?

2. ## 1st problem

Finally I solved the first one. I thought $X_n \Rightarrow X$ (in distribution) means that $F_{X_n}(t) \rightarrow F_X(t)$ FOR ALL t.

While it is the real definition is $F_{X_n}(t) \rightarrow F_X(t)$ FOR ALL t WHERE F_X is CONTINUOUS.

So the solution is:

$F_{X}(t - \epsilon) = P\{X \le t - \epsilon \} =$
$= P\{X \le t-\epsilon,\quad X_n \le t \} + P\{X \le t-\epsilon, \quad X_n > t \}$
$\le P\{X_n \le t \} + P\{X \le t-\epsilon, \quad X_n > t \}$

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

Thus:
$\liminf_{n \rightarrow \infty}F_{X_n}(t) \ge F_{X}(t-\epsilon)$

Hence (FX is continuous in t): $\liminf_{n \rightarrow \infty}F_{X_n}(t) \ge F_{X}(t)$

Finally

$\limsup_{n \rightarrow \infty}F_{X_n}(t) \le F_{X}(t) \le \liminf_{n \rightarrow \infty}F_{X_n}(t)$

Which solves the problem!

The second one is still unsolved!