Usually we show that a random variable is normally distributed, but how to show a random variable is NOT normally distributed? For example, this game's strategy is to bet $\$1$ in the $(n+1)^{th}$ game if the total winning up to the $n^{th}$ game is $\le 0$, and we stop when the total winning is $>0$. How to show that $W_i = \sum_{i=1}^n A_i X_i$ does not have a normal distribution?
Suppose $X_1, X_2, \dots$ are independent random variables each with a $N(0, 1)$ distribution. Let $\mathcal F_n$ denote the information in $X_1, \dots, X_n$, and let $A_1, A_2, \dots$ be a sequence of random variables such that for each $j$, $A_j$ is $F_{j-1}$ measurable. Let $W_0 = 0$ and for $n > 0$.
Suppose that
$$A_n = \begin{cases} 1, \text{ if } (n = 1) \text{ or } (A_{n-1} = 1 \text{ and } W_{n-1} \le 0)\\ 0, \text{ otherwise} \end{cases} $$
How to show that for all $n>1$, $W_n = \sum_{i=1}^n A_i X_i$ does not have a normal distribution? What are its expected value $\mathbb E[W_n]$ and $P(W_n>0)$?