On the asymptotics of degree structure of configuration graphs with bounded number of edges

We consider configuration graphs with $N$ vertices. The degrees of vertices are independent identically distributed random variables having the power-law distribution with positive parameter $\tau$. We study properties of random graphs such that the sum of vertex degrees does not exceed $n$ and the parameter $\tau$ is a random variable uniformly distributed on the interval $[a,b], 0<a<b<\infty$. We find limit distributions of the number $\mu_r$ of vertices with degree $r$ for various types of variation of $N,n$ and $r$.