bibkey: “ferger2015empirical” authors: “Dietmar Ferger; Daniel Vogel” year: 2015 title: “Weak convergence of the empirical process and the rescaled empirical distribution function in the Skorokhod product space” doi: null url: “https://arxiv.org/abs/1506.04324v1” claim: “Theorem 2.1 gives independent limits for the global empirical process and a locally rescaled empirical distribution: a time-transformed Brownian bridge and a two-sided Poisson process.” strata_touched: [] license: “citation-only” triage: “anchor”

Global and local empirical limits

The checked primary version is arXiv:1506.04324v1, submitted 2015-06-13; its title page is dated 2015-06-16 and the PDF has 22 pages.

For iid observations with a fixed distribution function F, the paper considers the global empirical process scaled by the square root of sample size and the empirical mass in neighborhoods of width inverse sample size around a fixed point. The local process uses the left limit of F on the negative half-axis. Condition C.1, printed page 5, assumes the corresponding finite one-sided derivatives at that point; it permits a jump at the point.

Theorem 2.1, printed page 6, proves joint convergence in the product of two Skorokhod spaces. The limits are a Brownian bridge composed with F and a two-sided Poisson process whose rates are the one-sided derivatives. These limits are independent even though both statistics are computed from the same empirical distribution. This is literature-attested precedent for global/local asymptotic independence, not a new general principle of the parity model.

The theorem’s local scale has a finite limiting expected count. The parity output-budget calculation instead has a growing expected count of order q/sqrt(lambda), a changing signal distribution, possibly moving lattice phases, and actual dependent observations before a marked-count comparison. The source does not provide the calibrated compensated-score clock, the fixed-size posterior comparison, or exact minimax centering. Those hypotheses and transfers must be established within that model; a common “local process” description does not make the theorem directly applicable.

A uniform empirical bridge as an auxiliary marginal

The global marginal of Theorem 2.1, with a fixed uniform distribution, supplies the classical uniform empirical-process Brownian bridge. This marginal is enough for a direct rare-count decomposition: conditional on the total number of rare entries, their entry parameters are iid with a normalized distribution function. A uniform inverse-distribution representation expresses their process as the uniform empirical bridge composed with that distribution function. An independent binomial total-count fluctuation supplies the remaining normal component.

The present functional budget proof separately establishes uniform convergence of the changing normalized distribution functions to a continuous clock, a random-sample-size conditioning argument, and the actual-experiment transfer. It does not apply the source’s local Poisson limit to a regime with diverging local expected count. The arithmetic clock may have corners; continuity is the property needed for the composition step.