Optimized Regularity Estimates of Conditional Distribution of the Sample Mean

摘要

We prove an optimized estimate for the regularity of the conditional distribution of the empiric mean of a finite sample of IID random variables, conditional on the sample "fluctuations". Prior results, based on bounds in probability, provided a H?lder-type regularity of the conditional distribution. We establish a Lipschitz regularity, using bounds in expectation. The new estimate, extending a well-known property of Gaussian IID samples, is a crucial ingredient of the Multi-Scale Analysis of multi-particle Anderson-type random Hamiltonians in a Euclidean space. In particular, the H¨older regularity of the multi-particle eigenvalue distribution, sufficient for the localization analysis of N-particle lattice Hamiltonians, with N ≥ 3, needs to be replaced by Lipschitz regularity for similar Hamiltonians in the Euclidean space.