Neighbour-dependent point shifts and random exchange models: Invariance and attractors
Journal article, 2017

Consider a partition of the real line into intervals by the points of a stationary renewal point process. Subdivide the intervals in proportions given by i.i.d. random variables with distribution G supported by [0, 1]. We ask ourselves for what interval length distribution F and what division distribution G, the subdivision points themselves form a renewal process with the same F? An evident case is that of degenerate F and G. As we show, the only other possibility is when F is Gamma and G is Beta with related parameters. In particular, the process of 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.

Gamma distribution

random operator

neighbour-dependent shifts

random exchange

Dirichlet distribution

renewal process

Poisson process


adjustment process


Anton Muratov

Royal Institute of Technology (KTH)

Sergey Zuev

Chalmers, Mathematical Sciences, Analysis and Probability Theory

University of Gothenburg


1350-7265 (ISSN)

Vol. 23 1 539-551

Subject Categories

Probability Theory and Statistics



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