A Grid-Rate Condition for Valid Uniform Inference

Estimating a continuous functional $F: \X \to \R$ involves specifying $L_n^d$ nodes on $\X \subset \R^d$ for estimation and uniform inference. While asymptotically valid inference requires $L_n$ to increase with $n$, existing fixed-$L$ rules of thumb and heuristic data-driven approaches lack formal justification. This paper shows that, for functions within a Donsker class, the simple grid-growth condition $L_n=\omega(r_n^{1/4})$ is sufficient for valid inference for twice continuously differentiable functions estimable at the $r_n^{1/2}$ rate. This condition ensures that the approximation error is asymptotically negligible relative to the stochastic variation of the empirical process.