  We show that the distribution of large values of an additive function on the
integers, and the distribution of values of the additive function on the primes
are related to each other via a Levy Process. As a consequence we obtain a
converse to an old theorem of Halasz. Halasz proved that if f is an strongly
additive function with f (p) \in {0, 1}, then f is Poisson distributed on the
integers. We prove, conversely, that if f is Poisson distributed on the
integers then for most primes p, f(p) = o(1) or f(p) = 1 + o(1).
