In establishing weak convergence of von Mises' differentiable statistical functions to a normal distribution usually square integrability conditions with respect to the underlying kernel function are assumed. It is shown that these conditions can be weakened by assuming integrability of the von Mises' functional itself. In addition it is pointed out that in nontrivial cases the conditions of square integrability of the kernel do not hold whereas weak convergence of the von Mises' functional can still be proved.
"Note on Conditions for Weak Convergence of Von Mises' Differentiable Statistical Functions." Ann. Statist. 5 (2) 405 - 407, March, 1977. https://doi.org/10.1214/aos/1176343807