An Esséen-type inequality for probability density functions, with an application

In this note, an upper bound is provided for the supremum of the absolute value of the difference of the probability density functions of two k-dimensional random vectors. The bound involves integrals of the absolute value of the characteristic functions of the random vectors, and shares a general similarity with a bound obtained by Sadikova for distribution functions of two-dimensional random vectors. Sadikova's paper provided the impetus for this note. Special cases are considered, and an application is presented, regarding consistency of a kernel estimate in the framework of associated random variables.