Suppose we have $N$ i.i.d. random variables $X_1, \dotsc, X_N,$ with the extra twist that $N$ is itself a (positive) random variable, with mean $\mu_n \to \infty.$ (For example, take a sequence where $N = Poisson(\mu_n).$
Under which conditions is it true that
$$X = g(N) - f(N)\sum_{i=1}^N X_i$$ is asymptotically normal (and for what form of $f(N)?$ In the usual CLT, of course, $f(N)$ is $\sqrt{N},$ while $g(N)$ is $N \mu(X_i).$