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): = ∫ xlog x—is considered. A direct proof is given that I has the Kadec property in L₁—if yₙ converges weakly to y and I(yₙ ) converges to I( y), then yₙ 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.