*Tid:* **3 oktober 2016 kl 15.15-16.15.**
**Seminarierummet 3721**, Institutionen för
matematik, KTH, Lindstedtsvägen 25, plan 7.
Karta!
*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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