English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We consider $U$-statistics on row-column exchangeable matrices. We derive adecomposition for them, based on orthogonal projections on probability spacesgenerated by sets of Aldous-Hoover-Kallenberg variables. The specificity ofthese sets is that they are indexed by bipartite graphs, which allows for theuse of concepts from graph theory to describe this decomposition. Thedecomposition is used to investigate the asymptotic behavior of $U$-statisticsof row-column exchangeable matrices, including in degenerate cases. Inparticular, it depends only on a few terms of the decomposition, correspondingto the non-zero elements that are indexed by the smallest graphs, namedprincipal support graphs, after an analogous concept suggested by Janson andNowicki (1991). Hence, we show that the asymptotic behavior of a $U$-statisticand its degeneracy are characterized by the properties of their principalsupport graphs. Indeed, their number of nodes gives the convergence rate of a$U$-statistic to its limit distribution. Specifically, the latter is degenerateif and only if this number is strictly greater than 1. Also, when the principalsupport graphs are connected, we find that the limit distribution is Gaussian,even in degenerate cases.

Characterization of the asymptotic behavior of $U$-statistics on row-column exchangeable matrices