Let $x_1, x_2, x_3, …$ be i.i.d random variables with probability $1/2$ equal to $+1$ or $-1$. We know that rescaled partial sums converge in distribution to normal r.v.
$$\frac{1}{\sqrt{n}}\sum_{i=1}^{n} x_i \overset{d}{\to} \mathcal{N}(0,1).$$
Now assume that we are given i.i.d gaussian random variables $a_1, a_2, a_3, … \sim \mathcal{N}(0, 1)$ (independent from $(x_1, x_2, x_3, …)$), is it true that for almost all sequences $(a_1, a_2, …)$ we have
$$\frac{1}{\sqrt{n}}\sum_{i=1}^{n} a_ix_i \overset{d}{\to} \mathcal{N}(0,\sigma^{2}) \;?$$
Obviously, there are sequences $a_1, a_2, a_3, …$ where the convergence in distribution holds (e.g. $a_i = 1$) and where it doesn't hold (e.g. $a_i = 2^{i}$), but is it true that it hold almost surely? Note that I consider case with $\sigma^{2} = 0$ also as normal distribution (given by delta measure $\delta_{0}$).
I tried Lindeberg-Feller CLT but it seems that its conditions do not hold almost surely. We can also consider $a_i$ following some arbitrary distribution $D$ with finite moments.
Best Answer
Consider $S\equiv \{(b_k) \in \mathbb R^{\infty}: \frac 1 n \sum\limits_{k=1}^{n} b_k^{2}\to 1\}$. By the ordinary CLT we get $\frac 1 {\sqrt n} \sum\limits_{k=1}^{n} b_kx_k \to N(0,1)$in distribution for any $(b_k) \in S$. By Strong Law of Large Numbers $(a_i) \in S$ with probability $1$. Hence, the result is true.