Let $X$ be a random variable and $Y_i\sim N(0,1)$ be indepedent random viriables (also independent with $X$) where $N(0,1)$ is the normal distribution. Let $Z_i=X*Y_i$, then generally $Z_i$s are dependent but identically distributed.
Now consider $W_n=\frac 1 {\sqrt{n}\sigma}\sum_{i=1}^n (Z_i-\mu)$ where $\mu,\sigma^2$ are the mean and variance of $Z_i$. Will $W_n$ converge to a normal distribution $N(0,1)$?
A simple example is that when $X$ is a constant, in which case $W_n$s converge to a normal distribution in a straightforward manner. However, in general case I can neither come up with a proof nor a counter example. If such conclusion is wrong, can we add some restrictions on $X$ to make central limit theorem satisfy in this case? Thank you very much!
In general, no. For example, if $X\sim N(0,1)$ then for every $n$ your $W_n$ has the distribution of the product of a pair of independent $N(0,1)$ variables, which itself is not Gaussian.