Density estimation from an individual numerical sequence

This paper considers estimation of a univariate density from an individual numerical sequence. It is assumed that (1) the limiting relative frequencies of the numerical sequence are governed by an unknown density, and (2) there is a known upper bound for the variation of the density on an increasing sequence of intervals. A simple estimation scheme is proposed, and is shown to be L₁ consistent when (1) and (2) apply. In addition, it is shown that there is no consistent estimation scheme for the set of individual sequences satisfying only condition (1)