Mathematische Zeitschrift The Markov moment problem and de Finetti’s theorem: Part I

The Markov moment problem is to characterize sequences s0, s1, s2, . . . admitting the representation sn = ∫ 1 0 x f (x) dx, where f (x) is a probability density on [0, 1] and 0 ≤ f (x) ≤ c for almost all x. There are well-known characterizations through complex systems of non-linear inequalities on {sn}. Necessary and sufficient linear conditions are the following: s0 = 1, and 0 ≤ (−1)n−j ( n j ) n−j sj ≤ c/(n+ 1) for all n = 0, 1, . . . and j = 0, 1, . . . , n. Here, is the forward difference operator. This result is due to Hausdorff. We give a new proof with some ancillary results, for example, characterizing monotone densities. Then we make the connection to de Finetti’s theorem, with characterizations of the mixing measure.