Neighbour-dependent point shifts and random exchange models: invariance and attractors

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.