  We investigate conditions in order to decide whether a given sequence of real
numbers represents expected maxima or expected ranges. The main result provides
a novel necessary and sufficient condition, relating an expected maxima
sequence to a translation of a Bernstein function through its
L\'{e}vy-Khintchine representation.
  Key words and phrases: expected maxima; expected ranges; Bernstein functions,
L\'{e}vy-Khintchine representation, order statistics.
