Let $X_{i},i\in \mathbb{N}$ be a sequence of independent standard normal random variables, $\mathscr{F}_{n}$ the associated natural filtration, and $S_{n}=\sum_{i=1}^{n}X_{i}$. You are allowed to use that for a standard normal random variable we have $\mathbb{E}(X^{2p})=(2p-1)!!=\mathbb{\Pi}^{p}_{i=1}(2i-1)$.
(a) Show that there exists a constant $C_{p}$ such that for all integers $p\geq1$ $$\mathbb{E}((\max_{1\geq k\geq n}S_{k})^{2p})\leq C_{p}n^{p}$$
This constant I found by using Doob's optimal inequality theorem and $C_{p}$ is equal to $(\frac{2p}{2p-1})^{2p}(\frac{2}{\pi})^{p}$.
(b) Let $a_{n},n\geq0$ be a sequence of non-negative numbers such that $\sum_{n}a_{n}^{2}<\infty$. Show that infinite series
$$\sum_{n=1}^{\infty}a_{n}X_{n}$$
converges in $L^{2}$ and almost surely.
(c) Show that the martingale
$$M_{n}:=\exp(S_{n}-\frac{1}{2}n)$$
converges almost surely, but not in $L^{1}$.
b) is starightforward; just show that partial sums form a Cauchy sequnce in $L^{2}$. For c) use SLLN to conclude that $e^{(S_n-\frac n 2)}=e^{n(\frac {S_n} n-\frac 1 2)}\to 0$ almost surely. Since $Ee^{(S_n-\frac n 2)}=(e^{\frac 1 2 })^{n} e^{-\frac n 2}=1$ for all $n$ we cannot have $L^{1}$ convergence.