Extremal properties of some geometric processes

In this paper some isoperimetric inequalities for stationary random tessellations are discussed. At first, classical results on deterministic tessellations in the Euclidean plane are extended to the case of random tessellations. An isoperimetric inequality for the random Poisson polygon is derived as a consequence of a theorem of Davidson concerning an extremal property of tessellations generated by random lines inR ². We mention extremal properties of stationary hyperplane tessellations inR ^d related to Davidson's result in cased=2. Finally, similar problems for random arrangements ofr-flats inR ^d are considered (r).This work was done while the author was visiting the University of Strathclyde in Glasgow.     

Shape-driven nested Markov tessellations

A new and rather broad class of stationary random tessellations of the d-dimensional Euclidean space is introduced, which we call shape-driven nested Markov tessellations. Locally, these tessellations are constructed by means of a spatio-temporal random recursive split dynamics governed by a family of Markovian split kernel, generalizing thereby the – by now classical – construction of iteration stable random tessellations. By providing an explicit global construction of the tessellations, it is shown that under suitable assumptions on the split kernels (shape-driven), there exists a unique time-consistent whole-space tessellation-valued Markov process of stationary random tessellations compatible with the given split kernels. Beside the existence and uniqueness result, the typical cell and some aspects of the first-order geometry of these tessellations are in the focus of our discussion.     

How many T‐tessellations on k lines? Existence of associated Gibbs measures on bounded convex domains

The paper bounds the number of tessellations with T‐shaped vertices on a fixed set of k lines: tessellations are efficiently encoded, and algorithms retrieve them, proving injectivity. This yields existence of a completely random T‐tessellation, as defined by Kiêu et al. (Spat Stat 6 (2013) 118–138), and of its Gibbsian modifications. The combinatorial bound is sharp, but likely pessimistic in typical cases. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 561–587, 2015     

Length distributions of edges in planar stationary and isotropic STIT tessellations

Stationary and isotropic random tessellations of the euclidean plane are studied which have the characteristic property to be stable with respect to iteration (or nesting), STIT for short. Since their cells are not in a face-to-face position, three different types of linear segments appear. For all the types the distribution of the length of the typical segment is given. The text was submitted by the authors in English.     

The Combinatorial Structure of Spatial STIT Tessellations

Spatially homogeneous random tessellations that are stable under iteration (nesting) in the $3$ 3 -dimensional Euclidean space are considered, so-called STIT tessellations. They arise as outcome of a space-time process of subsequent cell division and, consequently, they are not facet-to-facet. The intent of this paper is to develop a detailed analysis of the combinatorial structure of such tessellations and to determine a number of new geometric mean values, for example for the neighbourhood of the typical vertex. The heart of the results is a fine classification of tessellation edges based on the type of their endpoints or on the equality relationship with other types of line segments. In the background of the proofs are delicate distributional properties of spatial STIT tessellations.     

On Generalized Planar Random Tesellations

The mean value formulae of MECKE for planar random tessellations are true also for tessellations with not-necessarily convex cells. The same is true for a formula of Ambartzumian for the mean of the product of area and perimeter length of the “typical” cell. While the mean area of the cell containing the origin is greater than that of the “typical” cell, for mean perimeter length and mean edge number analogous inequalities are not true in general.     

Mixed Random Mosaics

Cowan [2] has defined random mosaics processes RMP in R² and has given some basic properties of them. In particular Cowan introduces the fundamental parameters α, β_k, γ_k of the process and, in terms of them, he computes the mean values of the area α, perimeter h, number of ares w and number of vertices v of a typical polygon of the RMP. Our purpose is to consider the RMP obtained by superposition of two independent random mosaics. Then, the characteristics a₁₂, h₁₂, w₁₂, v₁₂ of the resulting process are computed in terms of the characteristics a_i, h_i, w_i, v_i, of each process. The case of non random tessellations mixed with random mosaics is also considered.     

On the distribution of the longest run in number partitions

We consider the distribution of the longest run of equal elements in number partitions (equivalently, the distribution of the largest gap between subsequent elements); in a recent paper, Mutafchiev proved that the distribution of this random variable (appropriately rescaled) converges weakly. The corresponding distribution function is closely related to the generating function for number partitions. In this paper, this problem is considered in more detail—we study the behavior at the tails (especially the case that the longest run is comparatively small) and extend the asymptotics for the distribution function to the entire interval of possible values. Additionally, we prove a local limit theorem within a suitable region, i.e. when the longest run attains its typical order n ^1/2, and we observe another phase transition that occurs when the largest gap is of order n ^1/4: there, the conditional probability that the longest run has length d, given that it is ≤d, jumps from 1 to 0. Asymptotics for the mean and variance follow immediately from our considerations.     

The asymptotic distribution of long cycles in random regular graphs

The asymptotic distribution of the number of cycles of length l in a random r‐regular graph is determined. The length of the cycles is defined as a function of the number of vertices n, thus l=l(n), and the length satisfies l(n)→∞ as n→∞. The limiting distribution turns out to depend on whether l(n)/n→0 or l(n)/n→q, 0<q<1 as n→∞. In the first case the limit distribution is a weighted sum of Poisson variables while in the other case the limit distribution is similar to the limit distribution of Hamiltonian cycles in a random r‐regular graph. ©1999 John Wiley & Sons, Inc. Random Struct. Alg., 15: 43–92, 1999     

Topological Mean Value Relations for Random Cell Complexes

There are investigated stationary random q-dimensional topological cell complexes in ?^d, in particular, random tessellations. General relationships between the mean values of topological characteristics are derived. Then they are specified for the cases d = 2, 3, 4.     

On the exact distribution of linear combinations of order statistics from dependent random variables

We study the exact distribution of linear combinations of order statistics of arbitrary (absolutely continuous) dependent random variables. In particular, we examine the case where the random variables have a joint elliptically contoured distribution and the case where the random variables are exchangeable. We investigate also the particular L-statistics that simply yield a set of order statistics, and study their joint distribution. We present the application of our results to genetic selection problems, design of cellular phone receivers, and visual acuity. We give illustrative examples based on the multivariate normal and multivariate Student t distributions.     

On the internal path length of d‐dimensional quad trees

It is proved that the internal path length of a d‐dimensional quad tree after normalization converges in distribution. The limiting distribution is characterized as a fixed point of a random affine operator. We obtain convergence of all moments and of the Laplace transforms. The moments of the limiting distribution can be evaluated from the recursion and lead to first order asymptotics for the moments of the internal path lengths. The analysis is based on the contraction method. In the final part of the paper we state similar results for general split tree models if the expectation of the path length has a similar expansion as in the case of quad trees. This applies in particular to the m‐ary search trees. ©1999 John Wiley & Sons, Inc. Random Struct. Alg., 5: 25–41, 1999     

Reversible coagulation–fragmentation processes and random combinatorial structures: Asymptotics for the number of groups

The equilibrium distribution of a reversible coagulation‐fragmentation process (CFP) and the joint distribution of components of a random combinatorial structure (RCS) are given by the same probability measure on the set of partitions. We establish a central limit theorem for the number of groups (= components) in the case a(k) = qk^p?1, k ≥ 1, q, p > 0, where a(k), k ≥ 1, is the parameter function that induces the invariant measure. The result obtained is compared with the ones for logarithmic RCS's and for RCS's, corresponding to the case p < 0. © 2004 Wiley Periodicals, Inc. Random Struct. Alg. 2004     

A class of multivariate skew-normal models

The existing model for multivariate skew normal data does not cohere with the joint distribution of a random sample from a univariate skew normal distribution. This incoherence causes awkward interpretation for data analysis in practice, especially in the development of the sampling distribution theory. In this paper, we propose a refined model that is coherent with the joint distribution of the univariate skew normal random sample, for multivariate skew normal data. The proposed model extends and strengthens the multivariate skew model described in Azzalini (1985,Scandinavian Journal of Statistics,12, 171–178). We present a stochastic representation for the newly proposed model, and discuss a bivariate setting, which confirms that the newly proposed model is more plausible than the one given by Azzalini and Dalla Valle (1996,Biometrika,83, 715–726).     

An Iterative Approach to Determining the Length of the Longest Common Subsequence of Two Strings

This paper concerns the longest common subsequence (LCS) shared by two sequences (or strings) of length N, whose elements are chosen at random from a finite alphabet. The exact distribution and the expected value of the length of the LCS, k say, between two random sequences is still an open problem in applied probability. While the expected value E(N) of the length of the LCS of two random strings is known to lie within certain limits, the exact value of E(N) and the exact distribution are unknown. In this paper, we calculate the length of the LCS for all possible pairs of binary sequences from N=1 to 14. The length of the LCS and the Hamming distance are represented in color on two all-against-all arrays. An iterative approach is then introduced in which we determine the pairs of sequences whose LCS lengths increased by one upon the addition of one letter to each sequence. The pairs whose score did increase are shown in black and white on an array, which has an interesting fractal-like structure. As the sequence length increases, R(N) (the proportion of sequences whose score increased) approaches the Chvátal–Sankoff constant a _c (the proportionality constant for the linear growth of the expected length of the LCS with sequence length). We show that R(N) is converging more rapidly to a _c than E(N)/N.     

Simulation of typical Cox–Voronoi cells with a special regard to implementation tests

We consider stationary Poisson line processes in the Euclidean plane and analyze properties of Voronoi tessellations induced by Poisson point processes on these lines. In particular, we describe and test an algorithm for the simulation of typical cells of this class of Cox–Voronoi tessellations. Using random testing, we validate our algorithm by comparing theoretical values of functionals of the zero cell to simulated values obtained by our algorithm. Finally, we analyze geometric properties of the typical Cox–Voronoi cell and compare them to properties of the typical cell of other well-known classes of tessellations, especially Poisson–Voronoi tessellations. Our results can be applied to stochastic–geometric modelling of networks in telecommunication and life sciences, for example. The lines can then represent roads in urban road systems, blood arteries or filament structures in biological tissues or cells, while the points can be locations of telecommunication equipment or vesicles, respectively.     

Geometric Measures for Random Curved Mosaics of Rd

This paper deals with stationary random mosaics of R^d with general cell shapes. As geometric measures concentrated on the i-skeleton (i = 0, 1,…,d) the i-dimensional surface area (volume) measure and (i — 1) different curvature measures are chosen. The corresponding densities are calculated as well as for the mosaics and their superpositions in terms of mean cell parameters and mean cell numbers. This leads to various relations between the characteristic which are applied, in particular, to two- and three-dimensional tessellations. A comparison with known formulas for mosaics with convex cells in R² and R³ is given.     

Inhomogeneous random planar tessellations generated by lines

Random planar tessellations in bounded convex windows are generated by dividing random cells with random lines. It is suggested that the random STIT tessellations of Nagel and Weiss, if restricted to a bounded convex window, can be interpreted as a special case.     

Poisson polyhedra in high dimensions

The zero cell of a parametric class of random hyperplane tessellations depending on a distance exponent and an intensity parameter is investigated, as the space dimension tends to infinity. The model includes the zero cell of stationary and isotropic Poisson hyperplane tessellations as well as the typical cell of a stationary Poisson Voronoi tessellation as special cases. It is shown that asymptotically in the space dimension, with overwhelming probability these cells satisfy the hyperplane conjecture, if the distance exponent and the intensity parameter are suitably chosen dimension-dependent functions. Also the high dimensional limits of the mean number of faces are explored and the asymptotic behaviour of an isoperimetric ratio is analysed. In the background are new identities linking the f-vector of the zero cell to certain dual intrinsic volumes.