On the regularity of the conditional distribution of the sample mean

We show that the hypothesis of regularity of the conditional distribution of the empiric average of a finite sample of IID random variables, given all the sample "fluctuations", which appeared in our earlier manuscript |1] in the context of the eigenvalue concentration analysis for multi-particle random operators, is satisfied for a class of probability distributions with piecewise-constant or sufficiently smooth probability density. It extends the well-known property of Gausssian IID samples.