Threshold results for U-statistics of dependent binary variables

Large sample results for certain U-statistics, and related statistics, of b inary dependent random variables are studied. The class of U-statistics inc lude partial sums and polynomials of partial sums of a sequence of random v ariables. A very wide range of limit results are found. The form of the lim it result can depend substantially on the magnitude of the appropriate norm alizing sequence for the sum. Unexpectedly, the nature of the limit result also depends significantly on whether the degree of the U-statistic is even or odd. It is shown that dependence is a major factor contributing to this result. The limit results are illustrated with reference to a simple dynam ic sequence of binary variables and a reinforced random walk.