Optimal Wasserstein exponent for Poisson–Gaussian invariance

Determine the optimal exponent in the Wasserstein-distance bound comparing a multiple Poisson integral with the same-kernel multiple Gaussian integral in terms of the fourth add-one energy, beyond the sixth-root estimate established in Theorem 4 and the demonstrated failure of a square-root-order bound.

Background

The paper establishes a quantitative same-kernel Poisson–Gaussian invariance principle. For a fixed chaos order, if F=I_q^η(f) and G=I_q^W(f) have the same kernel and variance, the smooth-test discrepancy is controlled by the square root of the fourth add-one energy. Smoothing this estimate yields a Wasserstein bound with exponent 1/6 in the fourth add-one energy.

Proposition 17 shows that a Wasserstein estimate of the same square-root order as the smooth-test bound cannot hold uniformly, even at fixed positive variance. The authors therefore leave open the question of what the sharp Wasserstein exponent is between the available sixth-root upper bound and the excluded square-root rate.