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

TL;DR

This paper establishes a new $L^1$ law of large numbers for functions of i.i.d. variables shifted by a consistent estimator, even when the function becomes unbounded at zero, broadening the scope of uniform convergence results.

Contribution

It introduces a uniform $L^1$ law of large numbers for functions evaluated at data shifted by estimators, including cases where the function is unbounded at zero.

Findings

Proves uniform convergence under broad conditions.

Handles functions with singularities at zero.

Extends classical laws to more general functions.

Abstract

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.