Neighbour-dependent point shifts and random exchange models: Invariance and attractors
Artikel i vetenskaplig tidskrift, 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.

adjustment process

Dirichlet distribution

random operator

attractor

Gamma distribution

random exchange

renewal process

neighbour-dependent shifts

Poisson process

Författare

Anton Muratov

Kungliga Tekniska Högskolan (KTH)

Sergey Zuev

Göteborgs universitet

Chalmers, Matematiska vetenskaper, Analys och sannolikhetsteori

Bernoulli

1350-7265 (ISSN)

Vol. 23 1 539-551

Ämneskategorier

Sannolikhetsteori och statistik

DOI

10.3150/15-bej755

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Senast uppdaterat

2024-08-14