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

A. Muratov ; Sergei Zuyev (Institutionen för matematiska vetenskaper, Analys och sannolikhetsteori)
Bernoulli (1350-7265). Vol. 23 (2017), 1, p. 539-551.
[Artikel, refereegranskad vetenskaplig]

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.

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



Denna post skapades 2017-01-16.
CPL Pubid: 247092

 

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Institutioner (Chalmers)

Institutionen för matematiska vetenskaper, Analys och sannolikhetsteoriInstitutionen för matematiska vetenskaper, Analys och sannolikhetsteori (GU)

Ämnesområden

Sannolikhetsteori och statistik

Chalmers infrastruktur