Weak convergence of $U$-statistics on a row-column exchangeable matrix

$U$-statistics are used to estimate a population parameter by averaging a function on a subsample over all the subsamples of the population. In this paper, the population we are interested in is formed by the entries of a row-column exchangeable matrix. We consider $U$-statistics derived from functions of quadruplets, i.e. submatrices of size $2 \times 2$. We prove a weak convergence result for these $U$-statistics in the general case and we establish a Central Limit Theorem when the matrix is also dissociated. We shed further light on these results using the Aldous-Hoover representation theorem for row-column exchangeable random variables. Finally, to illustrate these results, we give examples of hypothesis testing for bipartite networks.