## Bernoulli

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

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

#### 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
Revised: June 2015
First available in Project Euclid: 27 September 2016

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

Digital Object Identifier
doi:10.3150/15-BEJ755

Mathematical Reviews number (MathSciNet)
MR3556783

Zentralblatt MATH identifier
1364.60066

#### 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

#### References

• [1] Bingham, N.H. and Maejima, M. (1985). Summability methods and almost sure convergence. Z. Wahrsch. Verw. Gebiete 68 383–392.
• [2] Daley, D.J. and Vere-Jones, D. (2008). An Introduction to the Theory of Point Processes. Vol. II: General Theory and Structure, 2nd ed. Probability and Its Applications (New York). New York: Springer.
• [3] Du, Q., Emelianenko, M. and Ju, L. (2006). Convergence of the Lloyd algorithm for computing centroidal Voronoi tessellations. SIAM J. Numer. Anal. 44 102–119.
• [4] Du, Q., Faber, V. and Gunzburger, M. (1999). Centroidal Voronoi tessellations: Applications and algorithms. SIAM Rev. 41 637–676.
• [5] Hasegawa, M. and Tanemura, M. (1976). On the pattern of space division by territories. Ann. Inst. Statist. Math. 28 509–519.
• [6] Holley, R. and Liggett, T.M. (1981). Generalized potlatch and smoothing processes. Z. Wahrsch. Verw. Gebiete 55 165–195.
• [7] Liggett, T.M. and Spitzer, F. (1981). Ergodic theorems for coupled random walks and other systems with locally interacting components. Z. Wahrsch. Verw. Gebiete 56 443–468.
• [8] Lukacs, E. (1955). A characterization of the gamma distribution. Ann. Math. Statist. 26 319–324.
• [9] McKinlay, S. (2014). A characterisation of transient random walks on stochastic matrices with Dirichlet distributed limits. J. Appl. Probab. 51 542–555.
• [10] Okabe, A., Boots, B., Sugihara, K. and Chiu, S.N. (2000). Spatial Tessellations: Concepts and Applications of Voronoi Diagrams, 2nd ed. Wiley Series in Probability and Statistics. Chichester: Wiley.
• [11] Peccati, G. and Reitzner, M., eds. (2016). Stochastic Analysis for Poisson Point Processes: Malliavin Calculus, Wiener–Itô Chaos Expansions and Stochastic Geometry. Berlin: Springer.