StrongL 1-norm consistency of data based histogram estimates of densities

In this paper we investigate the problem of estimating thed-dimensional probability densityf(x),x∈R ^d from a sample of sizen. The non-parametric estimator is the data based histogramf _n (x) as defined in (1). Under suitable conditions, we have proved theL ₁-norm consistance of this estimate, that is $$\mathop {\lim }\limits_{n \to \infty } \int_{R^a } {|f(x) - f_n (x)|dx = 0,{\text{ a}}{\text{.s}}{\text{.}}}$$