Moments of convex distribution functions and completely alternating   sequences
  We solve the moment problem for convex distribution functions on $[0,1]$ in terms of completely alternating sequences. This complements a recent solution of this problem by Diaconis and Freedman, and relates this work to the L\'{e}vy-Khintchine formula for the Laplace transform of a subordinator, and to regenerative composition structures. 
Alexander Gnedin, Jim Pitman