  Let $S$ be a multigraded polynomial ring such that the degree of each
variable is a unit vector; so $S$ is the homogeneous coordinate ring of a
product of projective spaces. In this setting, we characterize the formal
Laurent series which arise as Hilbert series of finitely generated $S$-modules.
  Also we provide necessary conditions for a formal Laurent series to be the
Hilbert series of a finitely generated module with a given depth. In the
bigraded case (corresponding to the product of two projective spaces), we
completely classify the Hilbert series of finitely generated modules of
positive depth.
