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 rowcolumn exchangeable matrix. We consider U -statistics derived from functions of quadruplets, i.e. submatrices of size 2 × 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.