Local limit theorems for one class of distributions in probabilistic combinatorics

Let a function $f(z)$ be decomposed into a power series with nonnegative coefficients which converges in a circle of positive radius $R.$ Let the distribution of the random variable $\xi_n$, $n\in\{1,2,\ldots\}$, be defined by the formula
$$P\{\xi_n=N\}=\frac{\mathrm{coeff}_{z^n}\left(\frac{\left(f(z)\right)^N}{N!}\right)}{\mathrm{coeff}_{z^n}\left(\exp(f(z))\right)},\,N=0,1,\ldots$$
for some $|z|<R$ (if the denominator is positive). Examples of appearance of such distributions in probabilistic combinatorics are given. Local theorems on asymptotical normality for distributions of $\xi_n$ are proved in two cases: a) if $ f(z) = (1-z)^{-\la}, \, \la = \mathrm {const} \in(0,1]$ for $|z| <1$, and b) if all positive coefficients of expansion $ f (z) $ in a power series are equal to 1 and the set $A$ of their numbers has the form
$$ A = \{m^r \, | \, m \in \mathbb{N} \}, \, \, r = \mathrm {const},\; r \in \{2,3,\ldots\}.$$
A hypothetical general local limit normal theorem for random variables $ \xi_n$ is stated. Some examples of validity of the statement of this theorem are given.