I know that $(\gamma_i)$ is a square summable sequence and that $$ L_n :=\sum_{i=1}^n \gamma_i (W_i^2-1) \quad \quad \forall n\in \mathbb{N} $$ where $(W_i)$ is an i.i.d. sequence of std. normal distributed rvs.
I know that $(L_n)$ converges almost surely (Khinchin-Kolmogorov) towards $\sum_{i=1}^\infty \gamma_i (W_i^2-1)\in \mathcal{L}^2$.
One can derive that $$ Var \left( \sum_{i=1}^\infty \gamma_i (W_i^2-1) \right) = \sum_{i=1}^\infty \gamma_i^2 Var (W_i^2-1) = 2 \sum_{i=1}^\infty \gamma_i^2 < \infty. $$ and we especially know that it has mean.
Problem how do i derive the expectation of the limit?
I know that
$$
E\left( \sum_{i=1}^\infty \gamma_i (W_i^2-1) \right) = \sum_{i=1}^\infty E\gamma_i (W_i^2-1) =0
$$
if $\sum_{i=1}^\infty E|\gamma_i (W_i^2-1)|=\sum_{i=1}^\infty |\gamma_i| \, E| (W_i^2-1)|=c\sum_{i=1}^\infty |\gamma_i| \, <\infty$, but i don't have the absolute convergence, only square summability.