Htykuut

Let $X_{i},iin mathbb{N}$ 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 $pgeq1$
$$mathbb{E}((max_{1geq kgeq n}S_{k})^{2p})leq C_{p}n^{p}$$

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 $a_{n},ngeq0$ 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.