Let's say you have $X_1,...,X_n$ observations of a RV X, which is distributed according to some arbitrary prob. function. Further there is a deterministic function $f(X)=Z$,
$f: X \rightarrow [-1,1]$
Under what conditions is the CLT applicable to the function of a random variable?
In general, if we have a sequence of iid variables $X_i$, a sufficient condition for applying the CLT (convergence of the normalized sum to a gaussian) is that the variance of $X_i$ is finite. Here, we don't know about that, so we can't apply the CLT to $X_i$. Nevertheless, we want to apply the CLT not to $X_i$ but to $Z_i=f(X_i)$, where $f$ is arbitrary deterministic function (perhaps not continuous, perhaps very ugly) but with bounded codomain. This is the key. This means that the image of the function (and hence the values of $Z_i$) are bounded (included in the interval $[-1,1]$). And this implies that the variance of $Z_i$ is finite. Hence we can apply the CLT to $Z_i$ (and the law of large numbers too).