Consider $X_{n}$ independent random variables with normal distribution with parameters $\mu_{n} , \delta_{n}$ such as : $\sum \mu_{n}$ and $\sum \delta_{n}^{2}$ both converge. Now we need to prove that $\sum_{n} X_{n}$ converges in $L_{p}$.
As I thought we should use Cauchy criteria for convergence , so : $\mathbb{E}\left|\sum_{n<k<m} X_{k}\right|^{p}$ should tends to zero. As I know sum of random variables with normal distribution is random variable with normal distribution, so we need connect it with convergence of sum of parameters. But how? Any hints ?
Let $1\leq n <m$ and consider $Y \equiv \sum_n^{m} X_j$. Then $Y$ is normal with mean $ \sum_n^{m} \mu_j$ and variance $ \sum_n^{m} \delta_j ^{2}$. If $Z$ has standard normal distribution then $Y$ has same distribution as $ (\sum_n^{m} \mu_j) + (\sum_n^{m} \delta_j^{2})^{1/2} Z$. Hence $||Y||_p \leq |\sum_n^{m} \mu_j| + (\sum_n^{m} \delta_j^{2})^{1/2} ||Z||_p$ Clearly this last quantity $\to 0$ so the partial sums of the series $\sum X_n$ form a Cauchy sequence in $L^{p}$. Since $L^{p}$ is complete we have finished the proof.