A uniform $L^1$ law of large numbers for functions of i.i.d. random variables that are translated by a consistent estimator

We develop a new $L^1$ law of large numbers where the $i$-th summand is given by a function $h(\cdot )$ evaluated at $X_i-{\theta }_n$, and where $\theta_n \circeq \theta_n(X_1,X_2,\ldots,X_n)$ is an estimator converging in probability to some parameter $\theta \in \mathbb{R}$. Under broad technical conditions, the convergence is shown to hold uniformly in the set of estimators interpolating between $\theta$ and another consistent estimator ${\theta }_n^{\star }$. Our main contribution is the treatment of the case where $\mathrm{\mid }h\mathrm{\mid }$ blows up at $0$, which is not covered by standard uniform laws of large numbers.