Local probabilities of randomly stopped sums of power law lattice random variables

Let $X_1$ and $N\ge 0$ be integer valued power law random variables. For a randomly stopped sum $S_N=X_1+\cdots +X_N$ of independent and identically distributed copies of $X_1$ we establish a first order asymptotics of the local probabilities $P(S_N=t)$ as $t\to +\mathrm{\infty }$. Using this result we show the $k^{-\delta }$, $0\le \delta \le 1$ scaling of the local clustering coefficient (of a randomly selected vertex of degree $k$) in a power law affiliation network.