Convergence of best entropy estimates

Given a finite number of moments of an unknown density ̅ x on a finite measure space, the best entropy estimate-that nonnegative density x with the given moments which minimizes the Boltzmann-Shannon entropy I(x):=∫ x log x-is considered. A direct proof is given that I has the Kadec property in L₁-if Y_n converges weakly to ̅y and I(y_n) converges to I( ̅y ), then y_nn converges to ̅y in norm. As a corollary, it is obtained that, as the number of given moments increases, the best entropy estimates converge in L₁ norm to the best entropy estimate of the limiting problem, which is simply ̅ x in the determined case. Furthermore, for classical moment problems on intervals with ̅ x strictly positive and sufficiently smooth, error bounds and uniform convergence are actually obtained.