# U-stat with random kernel

U-statistics assume that the kernel remain fixed. I wonder if theorems in u-stat still hold true when the kernel is random. For instance, I estimate the kernel $$h$$ using data. The estimated kernel is denoted by $$h_n$$ with $$h_n \xrightarrow{p} h.$$ So $$h_n$$ changes with sample size.

My guess is it can work if $$h$$ belongs to P-Donsker class and $$\mathbb{E}[(h_n-h)^2]\xrightarrow{p}0$$. I am not sure if my guess is correct.

This is certainly going to depend on which theorem you're talking about, and how you're trying to insert the randomness in $$h$$ into the theorem.
But you might be able to use results from $$U$$-processes, which can be used to show uniform convergence over all possible choices of $$h$$ (and hence for whichever random one you pick). An excellent resource for an overview, though surely there have been some new results in the last twenty years, is Decoupling: From Dependence to Independence by Victor H. de la Peña and Evarist Giné.