Dependency graphs, upper bounds on cumulants and singular graphons

Abstract: Consider a sum of random variables $S = \sum_{v \in V} A_v$. If the variables $A_v$ are weakly dependent, then it is well known that under mild assumptions, the distribution of $S$ is close to a normal distribution. The theory of dependency graphs enables one to make this statement precise. In this framework, we shall present new bounds on the cumulants of $S$, which enable one to have a combinatorial approach of this probabilistic results. One of the main application is the study of the fluctuations of the densities of subgraphs in a random graph chosen according to a graphon model. We shall see that two behavior are possible, according to whether the graphon is generic or singular. In the latter case, the limiting distributions that appear are non-Gaussian.

combinatorics

Audience: researchers in the topic

Warwick Combinatorics Seminar

Series comments: This is the online combinatorics seminar at Warwick.