Let $X_1, X_2,\dots$ be independent standard normal distributed random variables and $S_n=X_1+X_2+\dots+X_n$. For $a,b \in \mathbb{R}$ define
$$ Y_n=e^{aS_n-bn}.$$
- Show that $Y_n\xrightarrow{a.s.}0$ iff $b>0$.
- Show that $Y_n\xrightarrow{\mathcal{L}^p}0$ iff $b>\frac{a²p}{2}$.
I'm in need for some help on this exercise, I tried an approach with the Borel-Contelli-Lemma for the first task, but I am not sure if that's the right way to do it.
Thanks in advance!
Best Answer
Here are some hints:
For part 1: note that $\log Y_n = a S_n - bn = n (a\frac{S_n}{n} - b)$. By the strong law of large numbers, what does $a S_n / n -b$ converge to as $n$ goes to infinity?
For part 2: raise both sides to the $p$th power, and then use the fact that $$\mathbb{E}[e^{tX}] = e^{t^2 /2}$$ for a standard normal.