Asymptotic distributions for goodness-of-fit statistics in a sequence of multinomial models

We consider f-disparities between discrete distributions pn=(pn1,...,pnkn) and their estimates based on relative frequencies in an i.i.d. sample of size n, where is twice continuously differentiable in a neighborhood of 1 with f''(1)[not equal to]0. We derive asymptotic distributions of the disparity statistics under certain assumptions about pn and the second derivatives f'' in a neighborhood of 1. These assumptions are weaker than those known from the literature.