MAJORIZATION AND DIVERGENCE

A majorization ordering of the distance or divergence of a positive function from a fixed positive function q on a measure space is studied. If a function p1 is smaller than a function p2 relative to the ordering, then p1 is closer to q than p2. Applications are given to measures of divergence, Markov chains, and qualitative conditions for probability densities closest to q subject to first and second moment constraints. In addition, a generalization of the path result of R. F. Muirhead 1903, Proc. Edinburgh Math. Soc.21, 144-157) and G. H. Hardy, J. E. Littlewood, and G. Polya ("Inequalities," 2nd ed., Cambridge Univ. Press, London 1952) is given. © 1990.