Self-Normalized Moderate Deviations for Degenerate U-Statistics

In this paper, we study self-normalized moderate deviations for degenerate U-statistics of order 2. Let {, >= 1} be i.i.d. random variables and consider symmetric and degenerate kernel functions in the form h(,)=Sigma(infinity)(=1) ()(), where > 0, ((1)) = 0, and ((1)) is in the domain of attraction of a normal law for all >= 1. Under the condition Sigma(infinity)(=1) < infinity and some truncated conditions for {((1)): >= 1}, we show that log Sigma 1 <=not equal <= h(,)/max1 <=<infinity(2)(,) >= (2) similar to -(/2) for ->infinity and = (--root), where (2), = Sigma = 1(2)(). As application, a law of the iterated logarithm is also obtained.