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1 2001

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Layering of the Poisson process in the quadrant

Levikson B., Rolski T., Weiss G.

Probability and Mathematical Statistics

2001

Vol. 21, Fasc. 2

417--440

We consider the increasing sequence of non-intersecting monotone decreasing step processes Y*n(t), n = 1, 2,...(t > 0), whose jump points cover all the points of the homogeneous rate 1 Poisson process on the quadrant R2+. We deriveproperties of these processes, in particular the marginal distributions P(Y*n(t) > x), in terms of a Toeplitz determinant of some modified Bessel functions. Our system provides a new view of the Hammersley interacting particle system discussed by Aldousand Diaconis, and the distributions we derive are related tothe distribution of the length of the longest ascending sequence in a random permutation.