Abstract
An upper bound for the supremum of the absolute value of the difference of two multivariate probability density functions is obtained. The upper bound involves integrals of the absolute value of suitable transforms of the characteristic functions of the probability density functions. Results are similar to the work of Gamkrelidze (Theory Probab. Appl. 22 (1977) 877–880) on the Esseen's inequality for multidimensional distribution functions.
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