A mathematical preprint has derived finite-sample bounds for how closely the norm, or overall size, of a nondegenerate U-statistic can be approximated by a Gaussian counterpart. It measures the gap with Kolmogorov distance, the largest difference between two distribution functions, and bounds that gap by a quantity called Δ_n when the paper’s moment measure m_n is finite and its spectral factor ℓ_n is positive.
This is a methods result, not an analysis of an empirical population. The framework uses independent random elements that need not be identically distributed and places them in a separable Hilbert space H_n. In the Euclidean case H_n = R^{d_n}, the paper gives dimension-explicit Gaussian bounds under coordinatewise fourth-moment conditions.
Where the geometry enters
To make the comparison, the analysis uses an exact Hoeffding decomposition. It splits the normalized statistic into the Hájek projection L_n, a linear component, and a higher-order degenerate remainder R_n.
The size of the bound also depends on the covariance spectrum. The factor ℓ_n measures the proportion of squared spectral mass outside the leading eigendirection. That condition requires some mass beyond a single dominant direction: if the covariance is rank one, ℓ_n is zero, so the requirement ℓ_n > 0 is not met.
Under uniform moment and spectral conditions in a fixed separable Hilbert space, the Gaussian Kolmogorov bound has order n^-1/8. That exponent is a theoretical upper-bound rate for distributional error, not an error estimate measured from observations.
When the space grows as R^{d_n}, dimension becomes part of the calculation. The paper derives dimension-explicit Gaussian bounds under coordinatewise fourth-moment conditions, while the spectral behavior of the covariance remains part of the rate.
Bootstrap rates diverge
The study examines three bootstrap references: the empirical bootstrap, the Gaussian-weighted bootstrap and the jackknife multiplier bootstrap. Each is compared conditionally with the Gaussian target, asking how well the procedure reproduces the same norm distribution.
A general bootstrap theorem separates approximation error into three conceptual components. The detailed balance can include covariance-estimation terms and higher-order bootstrap remainders, with the contribution depending on the procedure.
In the fixed-space setting, the methods do not share one rate. The empirical bootstrap has an n^-1/8 rate, while the Gaussian-weighted and jackknife multiplier bootstraps each have an n^-1/6 rate under the paper’s balance, moment and spectral assumptions. These are derived rates, not results from a benchmark or data set.
For growing-dimensional Euclidean spaces, Condition B.3 makes the bootstrap-to-Gaussian error the same order, Δ_n,d, as the Gaussian approximation. Without that condition, covariance estimation can add a slower term.
From approximation to testing
The approximation results feed into norm-based tests. Under the settings of the stated propositions, the probability that the bootstrap test rejects a true null hypothesis converges to the nominal level α. This is an asymptotic calibration statement under assumptions, not an observed false-positive rate.
The power result is stated as a sufficient large-signal condition: consistency holds when ||θ||/ρ_n(Γ_n) tends to infinity. Here ρ_n(Γ_n) is the paper’s separation radius. The result does not establish local power exactly at the transition scale.
A separate calculation concerns the vector of pairwise Kendall’s tau coefficients in a specified dense Gaussian correlation model. Combined upper and lower bounds give a minimax separation rate of (p/n)^(1/2), up to multiplicative constants. The conclusion describes the best possible detection order within that model class, not a universal rate for every correlation problem.
The framework is also applied to vectorized Spearman correlation. Under conditions B.1 and B.2, its norm has a Gaussian approximation with a Δ_n,d-type bound, where the vector dimension is d = p(p - 1)/2. This is a theoretical application of the approximation result.
A theory paper, not an empirical test
The manuscript is an arXiv preprint, version 1, dated 26 Aug 2026. The supplied publication information does not report a journal or peer-review status.
The evidence is entirely theoretical: it consists of finite-sample inequalities, asymptotic testing consequences and minimax calculations under explicit assumptions. It does not provide an empirical sample, simulation study or direct finite-sample validation. The conclusions therefore apply to the stated Hilbert-space, spectral, moment, balance and model conditions.
Paper data and sources
Original title: Berry--Esseen bounds and bootstrap approximations for the Hilbert-space norm of $U$-statistics
Authors: Nilanjan Chakraborty, Sayan Das
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text