The almost sure invariance principles of degenerate U-statistics of degree two for stationary random variables

The almost sure invariance principle for some degenerate U-statistics of degree two is considered for strictly stationary random variables satisfying mixing conditions. The error term obtained is O(t1 - [gamma]) which yields Donsker's and Strassen's invariance principles for the U-statistics. The approach used here is based upon Hoeffding's expansion of kernel functions and the almost sure invariance principles for partial sums of Banach space valued mixing random variables.