Tid: 3 oktober 2016 kl 15.15-16.15.

Föredragshållare: Sergei Zuyev (Chalmers)

Titel: Segment recombinations and random sharing models

Abstract: Consider a renewal point process on the line and divide each of the segments it defines in proportion given by i.i.d. realisations of a fixed distribution supported by [0,1]. Now recombine the obtained pieces of the segments by joining the neighbouring ones, so that the division points are now the separation points between the new segments. We ask ourselves for which renewal processes and which division distributions the division points follow the same renewal process distribution? An evident case is that of equal length segments and a degenerate division distribution. Interestingly, the only other possible case is when the increments of the renewal process is Gamma and division points are Beta-distributed. In particular, the division points of a Poisson process is again Poisson, if the dividing distribution is Beta(r,1-r) for any r in (0,1). We show that a similar situation arises in the random sharing model when a countable number of `cites' exchange randomly distributed parts of their `wealth' with neighbours. More generally, Dirichlet distribution arises in these models as the distribution leading to a fixed point. We also show that the fixed points of the random sharing are attractors meaning that starting with a non-equilibrium configuration distribution will converge to the equilibrium. A joint work with Anton Muratov.

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