Consider a U-statistic of degree 2, based on a kernel h. Let h1(x) =

E[h(x, X2)] and ζ1 = V ar[h1(X1)]. Assume ζ1 > 0, so that we know that √n[Un −

θ(P)] converges in distribution to the normal distribution with mean 0 and variance 4ζ1. Consider estimating the limiting variance 4ζ1. Since Un averages h(Xi, X j) over the n 2 
pairs (Xi, X j) with i = j, one might use the sample variance of these n 2 
pairs as an estimator. That is, define S2 n = 1 n 2 
i< j [h(Xi, X j) − Un]
2 .
Determine whether or not S2 n is a consistent estimator. State any added conditions you might need. Generalize to U-statistics of degree b.
Section 12.4