Limit distributions of the number of vertices of a given degree in a configuration graph with bounded number of edges

We consider the model of an $N$-vertex configuration graph where the number of edges is at most $n$ and the degrees of vertices are independent and identically distributed (i.i.d.) random variables (r.v.'s). The distribution of the r.v. $\xi$, which is defined as the degree of any given vertex, is assumed to satisfy the condition $p_k=\mathbf{P}\{\xi=k\}\sim\frac{L}{k^g\ln^h k}$ as $k\to\infty$, where $L>0$, $g>1$, $h\ge0$. Limit theorems for the number of vertices of a given degree as $N, n\to\infty$ are proved.