In this note we show that almost sure convergence to zero of symmetrized U-statistics indexed by a linear sector in ℤd+ is equivalent to convergence along the diagonal of ℤd+, as it is considered in Latała and Zinn [Ann. Probab. 28 (2000) 1908–1924]. Comparisons with similar results for sums of multi-indexed i.i.d. random variables are also made.
"Sectorial convergence of U-statistics." Ann. Probab. 33 (2) 816 - 822, March 2005. https://doi.org/10.1214/009117904000001080