A uniform L1 law of large numbers for functions of i.i.d. random variables that are translated by a consistent estimator

We develop a new L1 law of large numbers where the ith summand is given by a function h(⋅) evaluated at Xi−θn, and where θn≗θn(X1,X2,…,Xn) is an estimator converging in probability to some parameter θ∈R. Under broad technical conditions, the convergence is shown to hold uniformly in the set of estimators interpolating between θ and another consistent estimator θn⋆. Our main contribution is the treatment of the case where |h| blows up at 0, which is not covered by standard uniform laws of large numbers.