Convergence sum of normal distributed random variables

convergence-divergencenormal distributionprobability theory

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
$$ Y_n=e^{aS_n-bn}.$$

  1. Show that $Y_n\xrightarrow{a.s.}0$ iff $b>0$.
  2. 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.

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