Almost sure \(L_ 1\)-norm convergence for data-based histogram density estimates (Q1819852)

Let F be a one-dimensional distribution function with density f, and let \(X_ 1,...,X_ n\) denote an i.i.d. sample drawn from F. It is shown that two conditions of a general nature on the random grids imply strong \(L_ 1\)-consistency of the data-based histogram density estimator \(f_ n(x)=f_ n(x;X_ 1,...,X_ n)\), i.e. \[ \int^{\infty}_{- \infty}| f_ n(x)-f(x)| dx\to 0\quad a.s.\quad as\quad n\to \infty. \] The generalization of the result to the multidimensional case is also indicated.