Distributionally scrambled set and minimal set

We investigate the relation between distributional chaos and minimal sets, and discuss how to obtain various distributionally scrambled sets by using least and simplest minimal sets. We show: i) an uncountable extremal distributionally scrambled set can appear in a system with just one simple minimal set: a periodic orbit with period 2; ii) an uncountable dense invariant distributionally scrambled set can occur in a system with just two minimal sets: a fixed point and an infinite minimal set; iii) infinitely many minimal sets are necessary to generate a uniform invariant distributionally scrambled set, and an uncountable dense extremal invariant distributionally scrambled set can be constructed by using just countably infinitely many periodic orbits.     

Discrete time Markov chains with interval probabilities

The parameters of Markov chain models are often not known precisely. Instead of ignoring this problem, a better way to cope with it is to incorporate the imprecision into the models. This has become possible with the development of models of imprecise probabilities, such as the interval probability model. In this paper we discuss some modelling approaches which range from simple probability intervals to the general interval probability models and further to the models allowing completely general convex sets of probabilities. The basic idea is that precisely known initial distributions and transition matrices are replaced by imprecise ones, which effectively means that sets of possible candidates are considered. Consequently, sets of possible results are obtained and represented using similar imprecise probability models.We first set up the model and then show how to perform calculations of the distributions corresponding to the consecutive steps of a Markov chain. We present several approaches to such calculations and compare them with respect to the accuracy of the results. Next we consider a generalisation of the concept of regularity and study the convergence of regular imprecise Markov chains. We also give some numerical examples to compare different approaches to calculations of the sets of probabilities.     

Diophantine properties of measures invariant with respect to the Gauss map

Motivated by the work of D. Y. Kleinbock, E. Lindenstrauss, G. A. Margulis, and B. Weiss [8, 9], we explore the Diophantine properties of probability measures invariant under the Gauss map. Specifically, we prove that every such measure which has finite Lyapunov exponent is extremal, i.e., gives zero measure to the set of very well approximable numbers. We show, on the other hand, that there exist examples where the Lyapunov exponent is infinite and the invariant measure is not extremal. Finally, we construct a family of Ahlfors regular measures and prove a Khinchine-type theorem for these measures. The series whose convergence or divergence is used to determine whether or not µ-almost every point is ψ-approximable is different from the series used for Lebesgue measure, so this theorem answers in the negative a question posed by Kleinbock, Lindenstrauss, and Weiss [8].     

Eliciting density ratio classes

The probability distributions of uncertain quantities needed for predictive modelling and decision support are frequently elicited from subject matter experts. However, experts are often uncertain about quantifying their beliefs using precise probability distributions. Therefore, it seems natural to describe their uncertain beliefs using sets of probability distributions. There are various possible structures, or classes, for defining set membership of continuous random variables. The Density Ratio Class has desirable properties, but there is no established procedure for eliciting this class. Thus, we propose a method for constructing Density Ratio Classes that builds on conventional quantile or probability elicitation, but allows the expert to state intervals for these quantities. Parametric shape functions, ideally also suggested by the expert, are then used to bound the nonparametric set of shapes of densities that belong to the class and are compatible with the stated intervals. This leads to a natural metric for the size of the class based on the ratio of the total areas under upper and lower bounding shape functions. This ratio will be determined by the characteristics of the shape functions, the scatter of the elicited values, and the explicit expert imprecision, as characterized by the width of the stated intervals. We provide some examples, both didactic and real, and conclude with recommendations for the further development and application of the Density Ratio Class.     

Some ergodic theorems for a parabolic Anderson model

In this paper,we study some ergodic theorems of a class of linear systems of interacting diffusions,which is a parabolic Anderson model.First,under the assumption that the transition kernel a=(a(i,j)) i,j∈s is doubly stochastic,we obtain the long-time convergence to an invariant probability measure νh starting from a bounded a-harmonic function h based on self-duality property,and then we show the convergence to the invariant probability measure νh holds for a broad class of initial distributions.Second,if(a(i,j)) i,j∈S is transient and symmetric,and the diffusion parameter c remains below a threshold,we are able to determine the set of extremal invariant probability measures with finite second moment.Finally,in the case that the transition kernel(a(i,j)) i,j∈S is doubly stochastic and satisfies Case I(see Case I in [Shiga,T.:An interacting system in population genetics.J.Math.Kyoto Univ.,20,213-242(1980)]),we show that this parabolic Anderson model locally dies out independent of the diffusion parameter c.     

Geometric ergodicity of Harris recurrent Marcov chains with applications to renewal theory

Let (X_n) be a positive recurrent Harris chain on a general state space, with invariant probability measure π. We give necessary and sufficient conditions for the geometric convergence of λPⁿf towards its limit π(f), and show that when such convergence happens it is, in fact, uniform over f and in L¹(π)-norm. As a corollary we obtain that, when (Xn) is geometrically ergodic, ∝ π(dx)6Pⁿ(x,·)-π6 converges to zero geometrically fast. We also characterize the geometric ergodicity of (X_n) in terms of hitting time distributions. We show that here the so-called small sets act like individual points of a countable state space chain. We give a test function criterion for geometric ergodicity and apply it to random walks on the positive half line. We apply these results to non-singular renewal processes on [0,∞) providing a probabilistic approach to the exponencial convergence of renewal measures.     

Inequalities for the Extremal Coefficients of Multivariate Extreme Value Distributions

The extremal coefficients are the natural dependence measures for multivariate extreme value distributions. For an m-variate distribution 2^m distinct extremal coefficients of different orders exist; they are closely linked and therefore a complete set of 2^m coefficients cannot take any arbitrary values. We give a full characterization of all the sets of extremal coefficients. To this end, we introduce a simple class of extreme value distributions that allows for a 1-1 mapping to the complete sets of extremal coefficients. We construct bounds that higher order extremal coefficients need to satisfy to be consistent with lower order extremal coefficients. These bounds are useful as lower order extremal coefficients are the most easily inferred from data.     

Isomorphism and Embedding of Borel Systems on Full Sets

A Borel system consists of a measurable automorphism of a standard Borel space. We consider Borel embeddings and isomorphisms between such systems modulo null sets, i.e. sets which have measure zero for every invariant probability measure. For every t>0 we show that in this category, up to isomorphism, there exists a unique free Borel system (Y,S) which is strictly t-universal in the sense that all invariant measures on Y have entropy <t, and if (X,T) is another free system obeying the same entropy condition then X embeds into Y off a null set. One gets a strictly t-universal system from mixing shifts of finite type of entropy ≥t by removing the periodic points and “restricting” to the part of the system of entropy <t. As a consequence, after removing their periodic points the systems in the following classes are completely classified by entropy up to Borel isomorphism off null sets: mixing shifts of finite type, mixing positive-recurrent countable state Markov chains, mixing sofic shifts, beta shifts, synchronized subshifts, and axiom-A diffeomorphisms. In particular any two equal-entropy systems from these classes are entropy conjugate in the sense of Buzzi, answering a question of Boyle, Buzzi and Gomez.     

Minimal ellipsoids and their duals

It is proved that “most” convex bodies in IE ^d touch the boundaries of their minimal circumscribed and their maximal inscribed ellipsoids in preciselyd(d+3)/2 points. A version of the former result shows that for “most” compact sets in IE ^d the corresponding optimal designs, i.e. probability measures with a certain extremal property, are concentrated ond(d+1)/2 points.     

Extremal functions as divisors for kernels of Toeplitz operators

In this paper, an extremal function of a Banach space of analytic functions in the unit disk (not all functions vanishing at 0) is a function solving the extremal problem for functions f of norm 1. We study extremal functions of kernels of Toeplitz operators on Hardy spaces H^p, 1<p<∞. Such kernels are special cases of so-called nearly invariant subspaces with respect to the backward shift, for which Hitt proved that when p=2, extremal functions act as isometric divisors. We show that the extremal function is still a contractive divisor when p<2 and an expansive divisor when p>2 (modulo p-dependent multiplicative constants). We give examples showing that the extremal function may fail to be a contractive divisor when p>2 and also fail to be an expansive divisor when p<2. We discuss to what extent these results characterize the Toeplitz operators via invariant subspaces for the backward shift.     

On linear forbidden submatrices

In this paper we study the extremal problem of finding how many 1 entries an n by n 0-1 matrix can have if it does not contain certain forbidden patterns as submatrices. We call the number of 1 entries of a 0-1 matrix its weight. The extremal function of a pattern is the maximum weight of an n by n 0-1 matrix that does not contain this pattern as a submatrix. We call a pattern (a 0-1 matrix) linear if its extremal function is O(n). Our main results are modest steps towards the elusive goal of characterizing linear patterns. We find novel ways to generate new linear patterns from known ones and use this to prove the linearity of some patterns. We also find the first minimal non-linear pattern of weight above 4. We also propose an infinite sequence of patterns that we conjecture to be minimal non-linear but have Ω(nlogn) as their extremal function. We prove a weaker statement only, namely that there are infinitely many minimal not quasi-linear patterns among the submatrices of these matrices. For the definition of these terms see below.     

The Return of the Baer Subplane

We discuss the notion of a tangency set in a projective plane, generalising the well-studied idea of a minimal blocking set. Tangency sets have recently been used in connection with the coding theory related to algebraic curves over finite fields, and they are closely related to the strong representative systems introduced by T. Illés, T. Szonyi, and F. Wettl (1991,Mitt. Math. Sem. Giessen201, 97–107). Here we give bounds on the possible sizes of tangency sets, and structural results are obtained in the extremal cases.     

Studies in possibilistic recognition

This paper introduces an algorithm for pattern recognition. The algorithm will classify a measured object as belonging to one of N known classes or none of the classes. The algorithm makes use of fuzzy techniques and possibility is used instead of probability. The algorithm was conceived with the idea of recognizing fast moving objects, but it is shown to be more general. Fuzzy ISODATA's use as a front end to the algorithm is shown. The algorithm is shown to accomplish the objectives of correct classification or no classification. Values that describe possibility distributions are introduced with some of their properties investigated and illustrated. An expected value for a possibility distribution is also investigated. The algorithm actually proves to be adaptable to a wide variety of imprecise recognition problems. Some test results illustrate the use of the technique embodied in the algorithm and indicate its viability.     

Markov Chains with finite convergence time

We study the properties of finite ergodic Markov Chains whose transition probability matrix P is singular. The results establish bounds on the convergence time of P^m to a matrix where all the rows are equal to the stationary distribution of P. The results suggest a simple rule for identifying the singular matrices which do not have a finite convergence time. We next study finite convergence to the stationary distribution independent of the initial distribution. The results establish the connection between the convergence time and the order of the minimal polynomial of the transition probability matrix. A queuing problem and a maintenance Markovian decision problem which possess the property of rapid convergence are presented.     

Embedding Toeplitz systems in triangular maps: The last but one problem of the Sharkovsky classification program

We give an example of a triangular map of the unit square containing a minimal Li–Yorke chaotic set and such that, in the whole system, there are no DC3-pairs. This solves the last but one problem of the Sharkovsky program of classification of triangular maps. We use completely new methods, in fact we show that every zero-dimensional almost 1–1 extension of the dyadic odometer can be realized as the unique nonperiodic minimal set in a triangular map of type 2^∞. In case of a regular Toeplitz system we can additionally arrange that all invariant measures are supported by minimal sets.     

Two testing problems relating the real and complex multivariate normal distributions

The first problem considered is that of testing for the reality of the covariance matrix of a p-dimensional complex normal distribution, while the second is that of testing that a 2p-dimensional real normal distribution has a p-dimensional complex structure. Both problems are reduced by invariance to their maximal invariant statistics, and the null and non-null distributions of these are obtained. Complete classes of unbiased, invariant tests are described for both problems, the locally most powerful invariant tests are obtained, and the admissibility of the likelihood ratio tests is established.     

Exterior Monge-Ampère solutions

We discuss the Siciak-Zaharjuta extremal function of a real convex body in Cn, a solution of the homogeneous complex Monge-Ampère equation on the exterior of the convex body. We determine several conditions under which a foliation by holomorphic curves can be found in the complement of the convex body along which the extremal function is harmonic. We study a variational problem for holomorphic disks in projective space passing through prescribed points at infinity. The extremal curves are all complex quadratic curves, and the geometry of such curves allows for the determination of the leaves of the foliation by simple geometric criteria. As a byproduct we encounter a new invariant of an exterior domain, the Robin indicatrix, which is in some cases the dual of the Kobayashi indicatrix for a bounded domain. Finally, we construct extremal curves for two non-convex bodies in R².     

Boundedness of the extremal solution for some p-Laplacian problems

We investigate the regularity of extremal solutions to some p-Laplacian Dirichlet problems with strong nonlinearities. Under adequate assumptions we prove the smoothness of the extremal solutions for some classes of nonlinearities. Our results suggest that the extremal solution’s boundedness for some range of dimensions depends on the nonlinearity f.