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

TL;DR

This paper investigates conditions under which certain renewal point processes and mass exchange models preserve their distributional properties, revealing invariance and attractor distributions like Gamma, Beta, and Dirichlet, with implications for Poisson processes.

Contribution

It characterizes when division points of renewal processes remain renewal processes with the same distribution and identifies fixed points and attractors in random exchange models.

Findings

Poisson process remains Poisson under Beta division.

Dirichlet distribution as a fixed point in exchange models.

Iterative Beta divisions lead to Poisson processes.

Abstract

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.