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On the number of reflexive and shared nearest neighbor pairs in one-dimensional uniform data

Bahadir S., Ceyhan E.

Probability and Mathematical Statistics

2018

Vol. 38, Fasc. 1

123--137

For a random sample of points in R, we consider the number, of pairs whose members are nearest neighbors (NNs) to each other and the, number of pairs sharing a common NN. The pairs of the first type are called, reflexive NNs, whereas the pairs of the latter type are called shared NNs. In, this article, we consider the case where the random sample of size n is from, the uniform distribution on an interval. We denote the number of reflexive NN pairs and the number of shared NN pairs in the sample by Rn and Qn, respectively. We derive the exact forms of the expected value and the variance for both Rn and Qn, and derive a recurrence relation for Rn which may also be used to compute the exact probability mass function (pmf) of Rn. Our approach is a novel method for finding the pmf of Rn and agrees with the results in the literature. We also present SLLN and CLT results for both Rn and Qn as n goes to infinity.

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.