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**Harvard**

Muratov, A. och Zuyev, S. (2017) *Neighbour-dependent point shifts and random exchange models: Invariance and attractors*.

** BibTeX **

@article{

Muratov2017,

author={Muratov, A. and Zuyev, Sergei},

title={Neighbour-dependent point shifts and random exchange models: Invariance and attractors},

journal={Bernoulli},

issn={1350-7265},

volume={23},

issue={1},

pages={539-551},

abstract={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.},

year={2017},

keywords={adjustment process, attractor, Dirichlet distribution, Gamma distribution, neighbour-dependent shifts, Poisson process, random exchange, random operator, renewal process},

}

** RefWorks **

RT Journal Article

SR Electronic

ID 247092

A1 Muratov, A.

A1 Zuyev, Sergei

T1 Neighbour-dependent point shifts and random exchange models: Invariance and attractors

YR 2017

JF Bernoulli

SN 1350-7265

VO 23

IS 1

SP 539

OP 551

AB 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.

LA eng

DO 10.3150/15-bej755

LK http://dx.doi.org/10.3150/15-bej755

OL 30