The LIL for canonical U-statistics of order 2

Let X, X-i, i epsilon N, be independent identically distributed random vari ables and let h(x, y) = h(y, x) be a measurable function of two variables. It is shown that the bounded law of the iterated logarithm, lim sup(n) log n(log n)(-1) \ Sigma1 less than or equal toi <j less than or equal ton h(X- i, X-j)\ < infinity a.s., holds if and only if the following three conditio ns are satisfied: h is canonical for the law of X [i.e., Eh(X, y) = 0 for a lmost all y] and there exists C < infinity such that both E(h(2)(X-1, X-2) boolean AND u) less than or equal to C log log u for all large u and sup(Eh (X-1, X-2) X f(X-1)g(X-2): \ \f(X)\ \ (2) less than or equal to 1, \ \g(x)\ \ (2) less than or equal to 1; \ \f \ \ (infinity) < <infinity>. \ \g \ \ (infinity) < infinity) less than or equal to C.