  We exploit results on the classical Stieltjes moment problem to obtain
completely explicit necessary and sufficient conditions for the photon number
distribution p(n) of a radiation field mode to be classical. These conditions
are given in two forms - respectively local and global in the individual photon
number probabilities. Central to the first approach is the recognition of the
important fact that the quantities n!p(n) are moments of a quasiprobability
distribution, notwithstanding the fact that p(n)'s can by themselves be
considered as a probability distribution over the nonnegative integers. This
leads to local classicality conditions involving p(n)'s for only a small number
of values of n. This local approach enables us to present detailed quantitative
statements on the connection between nonclassicality and oscillations in the
photon number distribution. The second approach is in terms of the traditional
factorial moments of p(n). Equivalence of the two approaches is established.
