Let's take a sequence $\{ a_n \}_{ n \in \mathbb{N} }$ in $l^2$, in other words assume that $\sum_{i = 0}^{\infty} a_i^2 < \infty$. If $\{ \xi_n \}_{ n \in \mathbb{N} }$ are i.i.d. $N(0,1)$, we know from Martingale theory that $\sum_{i = 0}^{n} a_i \xi$ converges a.s. Said that, I'd like to understand better the behavior of the squared sum.
What can we say about the sequence $ S_n = \sum_{i = 0}^{n} a_i^2 \xi_i^2$?
How can we prove or disprove its a.s. convergence?
The motivation is given from exercise 3.5, page 8, from these lecture notes (http://hairer.org/notes/SPDEs.pdf). If I understood correctly, we are basically asked to prove that $S_n$ converges almost surely, too, but my attempts failed. I tried with other Martingale strategies and direct inequalities, but I suspect the statement to be false and that I misunderstood the exercise. Thanks in advance.
Let $S_n:=\sum_{i=0}^na_i^2\xi_i^2$. Then for $m,n\geqslant 0$, $$ \mathbb E\left\lvert S_{m+n}-S_n\right\rvert=\sum_{i=n+1}^{n+m}a_i^2\mathbb E\left[\xi_i^2\right]=\sum_{i=n+1}^{n+m}a_i^2 $$ and consequently, the sequence $\left(S_n\right)_{n\geqslant 1}$ is Cauchy in $\mathbb L^1$. This sequence converges to a random variable $S$ in probability, and since $\left(S_n\right)_{n\geqslant 1}$ is non-decreasing, it also converges almost surely.