  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 $|h|$ blows up at $0$, which is
not covered by standard uniform laws of large numbers.
