$\lim\limits_{n\to\infty}\sum_{k=1}^n a_k X_k$ is Gaussian?

Question

If $X_k$, $k\in\mathbb N$ are Gaussian random variables and $a_k\in\mathbb R$, is it true that $\lim\limits_{n\to\infty}\sum_{k=1}^n a_k X_k$ is a gaussian random variable? I know that a finite linear combination of gaussian random variables is gaussian, but also the limit?

Answer 1

Set $a_k=1$ and $E[X] = 1$. Then $E[S_n] = n$ so $E[\lim S_n] = \lim n = \infty.$

Since normal variables have finite variance, $\lim S_n$ is not normal.

But, if, for instance, $a_k = (-1)^k$ or $E[X] =0$ the argument above would not hold. And, in particular, if $a_k = \frac{1}{n}$ CLT gives normality.