(a) State and prove the strong law of large numbers for sequences of i.i.d. random variables with a finite moment of order 4 .

(b) Let $\left(X_{k}\right)_{k \geqslant 1}$ be a sequence of independent random variables such that

$$\mathbb{P}\left(X_{k}=1\right)=\mathbb{P}\left(X_{k}=-1\right)=\frac{1}{2}$$

Let $\left(a_{k}\right)_{k \geqslant 1}$ be a sequence of real numbers such that

$$\sum_{k \geqslant 1} a_{k}^{2}<\infty$$

Set

$$S_{n}:=\sum_{k=1}^{n} a_{k} X_{k}$$

(i) Show that $S_{n}$ converges in $\mathbb{L}^{2}$ to a random variable $S$ as $n \rightarrow \infty$. Does it converge in $\mathbb{L}^{1}$ ? Does it converge in law?

(ii) Show that $\|S\|_{4} \leqslant 3^{1 / 4}\|S\|_{2}$.

(iii) Let $\left(Y_{k}\right)_{k \geqslant 1}$ be a sequence of i.i.d. standard Gaussian random variables, i.e. each $Y_{k}$ is distributed as $\mathcal{N}(0,1)$. Show that then $\sum_{k=1}^{n} a_{k} Y_{k}$ converges in law as $n \rightarrow \infty$ to a random variable and determine the law of the limit.