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

This paper characterizes the critical scaling frontier where shuffle privacy departs from Gaussian behavior. For binary randomized response with local privacy ε₀(n) satisfying eᵉ⁰/n → c², we prove experiment-level convergence (Le Cam distance) to explicit Poisson-shift and Skellam-shift limit experiments. For general finite alphabets in the sparse-error critical regime, we establish a multivariate compound-Poisson limit and an explicit limiting (ε,δ) privacy curve as a series. A three-regime phase diagram (sub-critical Gaussian, critical Poisson/Skellam, super-critical no-privacy) unifies the results. All bounds are explicit with O(1/n) rates.

Companion to arXiv:2602.09029.