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

TL;DR

This paper derives the asymptotic behavior of local probabilities for sums of power law lattice variables stopped at a random time, with applications to clustering coefficients in power law networks.

Contribution

It provides the first order asymptotics for local probabilities of randomly stopped sums of power law lattice variables and applies these results to network clustering analysis.

Findings

Asymptotic formula for $P(S_N=t)$ as $t o o

Power law scaling of local clustering coefficient with degree

Application to power law affiliation networks

Abstract

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 $t\to+\infty$. 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.