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Universal Shuffle Asymptotics, Part II: Non-Gaussian Limits for Shuffle Privacy -- Poisson, Skellam, and Compound-Poisson Regimes

Alex Shvets

TL;DR

The present paper characterizes the first universality-breaking frontier: critical sequences of increasingly concentrated local randomizers for which classical Lindeberg conditions fail and the shuffle score exhibits rare macroscopic jumps.

Abstract

Part I of this series (arXiv:2602.09029) develops a sharp Gaussian (LAN/GDP) limit theory for neighboring shuffle experiments when the local randomizer is fixed and has full support bounded away from zero. The present paper characterizes the first universality-breaking frontier: critical sequences of increasingly concentrated local randomizers for which classical Lindeberg conditions fail and the shuffle score exhibits rare macroscopic jumps. For shuffled binary randomized response with local privacy $\varepsilon_0 = \varepsilon_0(n)$, we prove experiment-level convergence (in Le Cam distance) to explicit shift limit experiments: a Poisson-shift limit for the canonical neighboring pair when $\exp(\varepsilon_0(n))/n \to c^2$, and a Skellam-shift limit for proportional compositions $k/n \to π\in (0,1)$ in the same scaling, including an explicit disappearance of the two-sided $δ$-floor away from boundary compositions. For general finite alphabets, we introduce a sparse-error critical regime and prove a multivariate compound-Poisson / independent Poisson vector limit for the centered released histogram, yielding a multivariate Poisson-shift experiment and an explicit limiting $(\varepsilon, δ)$ curve as a multivariate Poisson series. Together with Part I, these results yield a three-regime picture (Gaussian/GDP, critical Poisson/Skellam/compound-Poisson, and super-critical no privacy) under convergent macroscopic scalings.

Universal Shuffle Asymptotics, Part II: Non-Gaussian Limits for Shuffle Privacy -- Poisson, Skellam, and Compound-Poisson Regimes

TL;DR

The present paper characterizes the first universality-breaking frontier: critical sequences of increasingly concentrated local randomizers for which classical Lindeberg conditions fail and the shuffle score exhibits rare macroscopic jumps.

Abstract

Part I of this series (arXiv:2602.09029) develops a sharp Gaussian (LAN/GDP) limit theory for neighboring shuffle experiments when the local randomizer is fixed and has full support bounded away from zero. The present paper characterizes the first universality-breaking frontier: critical sequences of increasingly concentrated local randomizers for which classical Lindeberg conditions fail and the shuffle score exhibits rare macroscopic jumps. For shuffled binary randomized response with local privacy , we prove experiment-level convergence (in Le Cam distance) to explicit shift limit experiments: a Poisson-shift limit for the canonical neighboring pair when , and a Skellam-shift limit for proportional compositions in the same scaling, including an explicit disappearance of the two-sided -floor away from boundary compositions. For general finite alphabets, we introduce a sparse-error critical regime and prove a multivariate compound-Poisson / independent Poisson vector limit for the centered released histogram, yielding a multivariate Poisson-shift experiment and an explicit limiting curve as a multivariate Poisson series. Together with Part I, these results yield a three-regime picture (Gaussian/GDP, critical Poisson/Skellam/compound-Poisson, and super-critical no privacy) under convergent macroscopic scalings.
Paper Structure (27 sections, 33 theorems, 298 equations)

This paper contains 27 sections, 33 theorems, 298 equations.

Table of Contents

  1. Introduction
  2. Model and preliminaries
  3. Shuffle mechanism and transcript laws
  4. Privacy loss, privacy curves, and trade-off
  5. Total variation and Le Cam distance
  6. Binary randomized response and the scaling parameter
  7. Canonical neighboring pair: Poisson-shift limit
  8. Exact one-dimensional reduction
  9. Critical scaling and Poisson approximation
  10. Limiting privacy curve and the delta-floor
  11. Proportional compositions: Skellam-shift limit
  12. Centered decomposition
  13. Skellam limit
  14. Skellam pmf and limiting privacy curve
  15. General alphabets: multivariate compound-Poisson / independent Poisson vector limit
  16. ...and 12 more sections

Key Result

Lemma 2.1

Assume $Q \ll P$ and let $L = dQ/dP$. Then for every $\varepsilon \geq 0$, If the sample space is countable with pmfs $p$ and $q$, then for every $\varepsilon \geq 0$, The first identity requires $Q \ll P$; the second does not.

Theorems & Definitions (81)

  • Lemma 2.1: Neyman--Pearson identities for the privacy curve
  • proof
  • Lemma 2.2: Coupling bound for total variation
  • proof
  • Lemma 2.3: Contraction of total variation under measurable maps
  • proof
  • Lemma 2.4: Tensorization / union bound for independent pairs
  • proof
  • Lemma 2.5: Le Cam distance bounded by total variation on the same space
  • proof
  • ...and 71 more