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Universal Shuffle Asymptotics, Part III: Dominant-Block Quotient Geometry and Hybrid Gaussian--Compound-Poisson Limits in Finite-Alphabet Shuffle Privacy

Alex Shvets

Abstract

Part I of this series (arXiv:2602.09029) establishes a sharp Gaussian (LAN/GDP) limit theory for neighboring shuffle experiments in the fixed full-support regime. Part II (arXiv:2603.10073) identifies the first universality-breaking frontier: critical Poisson, Skellam, and multivariate compound-Poisson regimes. The present paper completes the finite-alphabet weak-limit theory by identifying the dominant-block quotient geometry that governs neighboring shuffle experiments. We treat dominant blocks of arbitrary finite size, allow overlap between the dominant output sets under the two neighboring hypotheses, and show that the limiting experiment decomposes according to this geometry: projecting onto the sum of the dominant tangent spaces yields a Gaussian factor, while quotienting by those same tangent spaces isolates a compound-Poisson jump field in the rare block. We also identify the regimes in which this quotient description determines the full privacy-curve, as well as the obstruction that appears when projected jump limits alone do not suffice. Two further sections sharpen the rate picture and the boundary interface: we show that the O(n^{-1/2}) rate for the full hybrid experiment is sharp in general, identify a compatibility condition restoring the O(n^{-1}) rate, and prove a boundary Berry--Esseen theorem giving O(c) Le Cam proximity between the critical Poisson-shift and Gaussian shift experiments as c tends to 0. Together with Parts I--II, this yields a three-regime universality picture and a precise finite-alphabet Levy--Khintchine layer for shuffle privacy.

Universal Shuffle Asymptotics, Part III: Dominant-Block Quotient Geometry and Hybrid Gaussian--Compound-Poisson Limits in Finite-Alphabet Shuffle Privacy

Abstract

Part I of this series (arXiv:2602.09029) establishes a sharp Gaussian (LAN/GDP) limit theory for neighboring shuffle experiments in the fixed full-support regime. Part II (arXiv:2603.10073) identifies the first universality-breaking frontier: critical Poisson, Skellam, and multivariate compound-Poisson regimes. The present paper completes the finite-alphabet weak-limit theory by identifying the dominant-block quotient geometry that governs neighboring shuffle experiments. We treat dominant blocks of arbitrary finite size, allow overlap between the dominant output sets under the two neighboring hypotheses, and show that the limiting experiment decomposes according to this geometry: projecting onto the sum of the dominant tangent spaces yields a Gaussian factor, while quotienting by those same tangent spaces isolates a compound-Poisson jump field in the rare block. We also identify the regimes in which this quotient description determines the full privacy-curve, as well as the obstruction that appears when projected jump limits alone do not suffice. Two further sections sharpen the rate picture and the boundary interface: we show that the O(n^{-1/2}) rate for the full hybrid experiment is sharp in general, identify a compatibility condition restoring the O(n^{-1}) rate, and prove a boundary Berry--Esseen theorem giving O(c) Le Cam proximity between the critical Poisson-shift and Gaussian shift experiments as c tends to 0. Together with Parts I--II, this yields a three-regime universality picture and a precise finite-alphabet Levy--Khintchine layer for shuffle privacy.
Paper Structure (13 sections, 21 theorems, 321 equations)

This paper contains 13 sections, 21 theorems, 321 equations.

Table of Contents

  1. Introduction
  2. Model and notation
  3. Shuffle mechanism, transcript laws, and privacy curves
  4. General finite-dominant regime and quotient geometry
  5. The general Lévy--Khintchine limit theorem
  6. Specializations.
  7. Proof of part (i): weak Lévy--Khintchine convergence
  8. Proof of part (ii): projected TV / Le Cam convergence
  9. Proof of part (iii): privacy-curve convergence
  10. The sharpness of order n to minus one-half in the hybrid experiment
  11. Boundary Berry--Esseen: Poisson/Skellam to Gaussian as c downarrow 0
  12. Discussion
  13. Technical lemmas

Key Result

Lemma 2.1

Let $\mu,\nu$ be probability measures on $(\mathcal{X},\mathcal{F})$ and let $f:\mathcal{X}\to\mathcal{Z}$ be measurable. Then

Theorems & Definitions (51)

  • Lemma 2.1: Contraction of total variation under measurable maps
  • proof
  • Lemma 2.2: Tensorization bound for product measures
  • proof
  • Lemma 2.3: Le Cam distance on the same space
  • proof
  • Lemma 2.4: Privacy-curve stability under total-variation perturbations
  • proof
  • Lemma 2.5: Data processing for privacy curves
  • proof
  • ...and 41 more