A sequence of random variables that does not converge in probability.

I was doing a problem about the converge of the sum of random variables which has two parts:

Let $X_1, X_2 ,\dots$ be independent and identically distributed random variables with $E X_i = 0$, $ 0 <\operatorname{Var}(X_i) < \infty $, and let $S_n = X_1 + \dots+ X_n$.

(a) Use the central limit theorem and Kolmogorov's zero-one law to conclude that $ \limsup S_n / \sqrt{n} = \infty$ almost surely.

(b) Use an argument by contradiction to show that $S_n / \sqrt{n}$ does not converge in probability. Hint: Consider $n = m! $.

I did part (a) but I'm not sure about my proof and people are welcome to go through it:

(a) Let $\operatorname{Var} (X_i) = \sigma ^2$, then by central limit theorem $\frac{S_n} {\sigma \sqrt{n}} \Rightarrow \chi$ where $\chi$ has the standard normal distribution. Therefore for any $x > 0$, $P( \limsup \frac{S_n}{\sigma \sqrt{n}} > x ) \ge P (\chi > x ) > 0$, thus $P ( \limsup \frac{S_n}{\sigma \sqrt{n}} > x) = 1$ for any $x>0$ by Kolmogorov's zero-one law. By monotonicity this implies $ P ( \limsup \frac {S_n}{\sqrt{n}} = \infty) =1 $, which is $\limsup \frac{S_n}{\sqrt{n}} = \infty$ a.s.

But I stuck with part (b), my approach is the following:

(b) Suppose $ \frac{ S_n}{\sqrt{n}}$ converges in probability, then the subsequence $\frac{S_{m!}}{\sqrt{m!}}$ has a further subsequence that converges almost surely.

But it doesn't seem to work out well and I don't think I can go on with it. How to prove part (b)? Any idea is appreciated.

Solution 1:

The first part looks ok, but I would apply central limit theorem, not the law of large number. The lower bound of the probability of the $\limsup$ has to be justified (portmanteau theorem).

For part b), we can use the following idea: in the case $S_n/\sqrt n\to \chi$ in probability, we would have $$Y_n:=\frac{S_{2n}}{\sqrt{2n}}-\frac{S_n}{\sqrt n}\to 0 \mbox{ in probability}.$$ But $Y_n=\frac{S_{2n}-S_n}{\sqrt{2n}}+\frac{S_n}{\sqrt n}\left(\frac 1{\sqrt 2}-1\right)=:Y'_n+Y''_n$. Since $S_{2n}-S_n$ is independent of $S_n$, we can compute the limit in distribution of each of the two terms which compose $Y_n$. Notice that $Y'_n$ has the same distribution as $S_n/\sqrt{2n}$ which converges in distribution to a centered normal random variable of variance $\sigma^2/2$, while $Y''_n$ converges in distribution to a centered normal random variable of variance $\sigma^2(1-\sqrt 2)^2/2$. Therefore, it can be shown that $Y_n$ converges to a non-degenerated Gaussian random variable.