  We study two rumor processes on $\N$, the dynamics of which are related to an
SI epidemic model with long range transmission. Both models start with one
spreader at site $0$ and ignorants at all the other sites of $\N$, but differ
by the transmission mechanism. In one model, the spreaders transmit the
information within a random distance on their right, and in the other the
ignorants take the information from a spreader within a random distance on
their left.
  We obtain the probability of survival, information on the distribution of the
range of the rumor and limit theorems for the proportion of spreaders. The key
step of our proofs is to show that, in each model, the position of the
spreaders on $\N$ can be related to a suitably chosen discrete renewal process.
