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This paper studies a practical problem: when you estimate a continuous curve from data, you usually compute estimates only at a finite set of grid points and then draw a continuous curve by connecting those points. The author shows a simple condition on how fast the number of grid points should grow with the sample size so that uniform confidence bands drawn from the interpolated curve are asymptotically valid. In plain terms: if the grid becomes dense fast enough, the error from interpolation becomes negligible compared with the estimator’s natural random variation.
To reach this conclusion the paper analyses linear and multilinear interpolation as the device that turns discrete estimates into a continuous curve. Under the assumption that the true function is twice continuously differentiable on a compact domain, the author derives a deterministic bound on the interpolation error. Concretely, the scaled interpolation error is bounded by a constant times sqrt(r_n) divided by (L−1)^2, where L is the number of nodes per axis and r_n is the rate at which the estimator’s variance shrinks (for usual parametric √n estimators, r_n = n). From this bound the paper shows that the interpolation error is asymptotically negligible whenever the grid satisfies L_n = ω(r_n^{1/4}) — that is, L_n must grow faster than r_n to the one‑quarter power.
The paper then studies the random part of the estimator under a Donsker condition. “Donsker class” is a technical assumption that ensures the estimator’s random fluctuations behave like a well‑behaved Gaussian process. Under this condition, and under the grid growth rule above, the author proves that the interpolated empirical process converges to the same Gaussian limit as the underlying continuous process. Theorem 3.1 shows the interpolated process inherits the same weak limit, so confidence bands formed after interpolation have the same asymptotic coverage properties as those defined for a truly continuous estimator. The paper also records a computational algorithm that makes the interpolation step explicit and mentions bootstrap implementations using weights with mean one and variance one.
This result matters because practitioners often face a trade‑off: too few grid points can miss important features between nodes and lead to invalid uniform inference, while too many points can be computationally expensive and cause finite‑sample problems. The paper gives a clear, rate‑based guideline that replaces ad hoc rules of thumb (for example, some prior advice suggested L_n = ω(n^{1/2}) or a fixed default like 99). The guideline is transparent: for estimators with the usual √n rate, it says L_n must grow faster than n^{1/4}.
There are important caveats. The main result assumes the target function is twice continuously differentiable; if the function is only Hölder continuous with exponent α (a weaker smoothness condition), the required grid rate changes to L_n = ω(r_n^{1/(2α)}). The theory also relies on the Donsker representation, which covers many semiparametric estimators (for example some distribution and quantile regression settings) but not all non‑Donsker situations. Finally, a very dense grid can still be computationally prohibitive in higher dimensions, and dense discretisation may create finite‑sample issues such as quantile‑crossing in quantile regression. The paper reports Monte Carlo evidence in support of the approach, with further simulation details in the supplement.