## Bernoulli

- Bernoulli
- Volume 23, Number 1 (2017), 539-551.

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

Anton Muratov and Sergei Zuyev

#### 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: $\mathrm{B} (r,1-r)$ 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 $\mathrm{B} (r,1-r)$-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.

#### Article information

**Source**

Bernoulli, Volume 23, Number 1 (2017), 539-551.

**Dates**

Received: September 2014

Revised: June 2015

First available in Project Euclid: 27 September 2016

**Permanent link to this document**

https://projecteuclid.org/euclid.bj/1475001365

**Digital Object Identifier**

doi:10.3150/15-BEJ755

**Mathematical Reviews number (MathSciNet)**

MR3556783

**Zentralblatt MATH identifier**

1364.60066

**Keywords**

adjustment process attractor Dirichlet distribution Gamma distribution neighbour-dependent shifts Poisson process random exchange random operator renewal process

#### Citation

Muratov, Anton; Zuyev, Sergei. Neighbour-dependent point shifts and random exchange models: Invariance and attractors. Bernoulli 23 (2017), no. 1, 539--551. doi:10.3150/15-BEJ755. https://projecteuclid.org/euclid.bj/1475001365