Optimized estimates of the regularity of the conditional distribution of the sample mean

We give an improved 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", extending the well-known property of Gaussian IID samples. Specifically, we replace the bounds in probability, established in our earlier works, by those in distribution, and this results in the optimal regularity exponent in the final estimate.