Let $(W_k)_{k \in \mathbb{Z}}$ be a iid sequence of $N(0,1)$ random variables and
$$X(f):=\sum_{k \in \mathbb{Z}}c_k(f)W_k, \ f \in L^2[-\pi, \pi]$$
with
$$c_k(f):=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)e^{-ikx}, \ k \in \mathbb{Z}$$
I'm trying to prove that this series converges in $L^2(\Omega, \mathcal{F}, P)$ and looking for hints. Is there any 'clever' way or trick to show this? I don't really know how to start.
I also want to determine $E[X(f)]$, $V[X(f)]$ and $Cov[X(f),X(g)]$ for $f,g \in L^2[-\pi, \pi]$
$E[X(f)]=\sum_{k \in \mathbb{Z}}c_k(f)E[W_k]=0$.
$V[X(f)]=\sum_{k \in \mathbb{Z}}(c_k(f))^2V[W_k]=\sum_{k \in \mathbb{Z}}(c_k(f))^2$
$Cov[X(f),X(g)]$=?
In the Hilbert space $L^{2}(P)$ the family $(W_k)$ is orthonormal. Hence $\sum a_kW_k$ converge sin $L^{2}(P)$ iff $\sum |a_k|^{2} <\infty$. In our case $\sum |c_f(k)|^{2} <\infty$ because $f \in L^{2} ([-\pi,\pi])$.