A note on the asymptotic law of the histogram without continuity assumptions

Abstract

Asymptotic normality of density estimates often requires the continuity of the underlying density and assumptions on its derivatives. Recently, these assumptions have been weakened for some estimates using the less restrictive notion of regularity index. However, the particular definition of this index makes it unusable for many estimates. In this paper, we define a more general regularity concept: the $r$-regularity. This concept is used to obtain asymptotic law of the histogram without hypothesis on the continuity of the underlying density. As expected, when it does exist, the limit distribution is a standard Gaussian. Then, to illustrate the new definition of $r$-regularity, examples are studied.