bibkey: “einmahl1997local” authors: “Uwe Einmahl; David M. Mason” year: 1997 title: “Gaussian approximation of local empirical processes indexed by functions” doi: “10.1007/s004400050086” url: “https://link.springer.com/content/pdf/10.1007/s004400050086.pdf” claim: “Theorem 1.1 gives weak convergence of local empirical processes under assumptions (A), (S), and (F), including shrinking local mass with diverging expected count.” strata_touched: [] license: “citation-only” triage: “anchor”

Growing-count local empirical Gaussian limits

Published in Probability Theory and Related Fields 107, 283–311 (1997). The checked primary publisher PDF has 29 pages. Definitions (1.1)–(1.3) and assumptions (A) are on printed page 284; assumptions (S) and (F) specify measurability, envelope integrability, convergence of localized distributions, and asymptotic equicontinuity. Theorem 1.1 is on printed page 287.

The observations are iid from a fixed distribution. Localization has mass a_n, with n a_n tending to infinity and a_n tending to a limit a. The local process is centered and divided by sqrt(n a_n). Equations (1.22)–(1.23) express the Gaussian limit as a localized Brownian bridge plus an independent normal component. Its covariance is P_0(fg)-a P_0(f)P_0(g). When a=0 it is P_0(fg); nested indicator sets therefore give Brownian covariance. This is a literature-attested predecessor for the growing-count Gaussian mechanism. The theorem’s regularity assumptions remain necessary to check when applying it; a matching covariance formula alone is not an application.

Chapter 28, Theorem 28.2 of the parity fluctuation volume is repo-derived: its substantive deductions are actual compensated-score calibration, control of arbitrary accepted subsets of a truncated lattice atom with separate tail remainders, the explicit phase clock, posterior true-count comparison, and exact finite minimax centering in both stationary experiments. Generic local Gaussian limits, likelihood-ratio ordering, ROC operating-point geometry and Brownian covariance are established ingredients, not new general theorems. The bounded primary-source search did not identify a theorem supplying all of those model-specific deductions; it does not certify global originality.

Expanding budgets and simultaneous scales

Theorem 30.2 of the parity fluctuation volume is repo-derived. Its new actual-model estimate sums total-count errors over dyadic output ranges using a posterior signal weight that decreases inversely with the output size. The sum is geometric. A single conditional concentration bound over all rank intervals transfers that estimate to the maximum hidden-label discrepancy, including its expectation and the exact finite minimax centers.

The local Gaussian empirical-process mechanism remains literature-attested. At the larger logarithmic band width, a direct compensated-score Berry–Esseen bound suffices to identify the clock; it would not identify the constant-width clock of Chapter 28. The growing normalized band has a phase-independent clock, whereas the narrower, differently normalized compact-budget field retains its arithmetic phase. Joint independence is proved from simultaneous row-vector covariances and tightness, not from a claim that separated scales are automatically independent or that parameter and sample-size limits may be interchanged.

The source’s growing-local-count theorem supplies the classical probability context, not the parity model’s dyadic posterior weighting, actual pair/path calibration or unknown-direction risk comparison. The bounded source review also examined local/tail empirical processes, intermediate quantile ranks, rejective sampling and recent tail-process work. It found no theorem supplying that entire actual-model transfer; this is not a global originality certificate.