On the Marcinkiewicz–Zygmund Strong Law for U-Statistics

Sen (when 0<p<1) and Giné and Zin (1<p<2) have given sufficient conditions for a Marcinkiewicz–Zygmund type SLLN to hold for U-statistics. As demonstrated by the latter authors, their requirement is not necessary when U is completely degenerate or 0<p<1. Here, for p in (1, 2), a simplified proof of sufficiency is obtained which weakens their integrability condition and in the nondegenerate case also extends the permissible values of p. This, in turn, leads to improved conditions for Hoeffding's CLT (likewise for the LIL) in the non-degenerate case. Moreover, nonnecessity is shown to hold in the SLLN when U is degenerate but not completely so.