  Consider a stationary renewal point process on the real line and divide each
of the segments it defines in a proportion given by \iid realisations of a
fixed distribution $G$ supported by [0,1]. We ask ourselves for which
interpoint distribution $F$ and which division distributions $G$, the division
points is again a renewal process with the same $F$? An evident case is that of
degenerate $F$ and $G$. Interestingly, the only other possibility is when $F$
is Gamma and $G$ is Beta with related parameters. In particular, the division
points of a Poisson process is again Poisson, if the division distribution is
Beta: B$(r,1-r)$ for some $0<r<1$. We show a similar behaviour of random
exchange models when a countable number of `agents' exchange randomly
distributed parts of their `masses' with neighbours. More generally, a
Dirichlet distribution arises in these models as a fixed point distribution
preserving independence of the masses at each step. We also show that for each
$G$ there is a unique attractor, a distribution of the infinite sequence of
masses, which is a fixed point of the random exchange and to which iterations
of a non-equilibrium configuration of masses converge weakly. In particular,
iteratively applying B$(r,1-r)$-divisions to a realisation of any renewal
process with finite second moment of $F$ yields a Poisson process of the same
intensity in the limit.
