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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.

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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.

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