 Variance as a U-statistic

The symmetric kernel $h(x,y)=(x-y{)}^2/2$ has expectation equal to the population variance for independent identically distributed observations with finite second moment. Its U-statistic reduces to the displayed unbiased sample variance, using ${\sum }_{i<j}(X_i-X_j{)}^2=n{\sum }_i(X_i-\overline{X}{)}^2$.