On Hoeffding decomposition in \(L_{p}\) (Q1928875)

The Hoeffding decomposition for a \(U\)-statistic of order \(d\) expresses the statistic as a sum of canonical statistics of orders \(k \leq d\). The classical result assumes the statistic is based on random variables with finite variance. \textit{J. Bourgain} [Semin. Anal. Fonct. 1979--1980, Expose No.4 (1980; Zbl 0454.46020)] extended the decomposition to the case with finite \(p\)th moments, \(p > 1\). The author provides a new approach to the proof of this result, improved \(L_p\) bounds, as well as new Rosenthal-type moment inequalities for canonical \(U\)-statistics.