Multilinear forms and measures of dependence between random variables

Based on an idea of Rosenblatt, the methods of interpolation theory are used to establish moment inequalities and equivalence relations for measures of dependence between two or more families of random variables. A couple of “interpolation” theorems proved here appear to be new.     

The net Bayes premium with dependence between the risk profiles

In Bayesian analysis it is usual to assume that the risk profiles Θ₁ and Θ₂ associated with the random variables “number of claims” and “amount of a single claim”, respectively, are independent. A few studies have addressed a model of this nature assuming some degree of dependence between the two random variables (and most of these studies include copulas). In this paper, we focus on the collective and Bayes net premiums for the aggregate amount of claims under a compound model assuming some degree of dependence between the random variables Θ₁ and Θ₂. The degree of dependence is modelled using the Sarmanov–Lee family of distributions [Sarmanov, O.V., 1966. Generalized normal correlation and two-dimensional Frechet classes. Doklady (Soviet Mathematics) 168, 596–599 and Ting-Lee, M.L., 1996. Properties and applications of the Sarmanov family of bivariate distributions. Communications Statistics: Theory and Methods 25 (6) 1207–1222], which allows us to study the impact of this assumption on the collective and Bayes net premiums. The results obtained show that a low degree of correlation produces Bayes premiums that are highly sensitive.     

CDS simulation and pattern formation of phase separation

Several properties of the generation and evolution of phase separating patterns for binary material studied by CDS model are proposed. The main conclusions are (1) for alloys spinodal decomposition, the conceptions of “macro-pattern” and “micropattern” are posed by “black-and- white graph” and “gray-scale graph” respectively. We find that though the four forms of map f that represent the self-evolution of order parameter in a cell (lattice) are similar to each other in “macro-pattern”, there are evident differences in their micro-pattern, e.g., some different fine netted structures in the black domain and the white domain are found by the micro-pattern, so that distinct mechanical and physical behaviors shall be obtained. (2) If the two constitutions of block copolymers are not symmetric (i.e. r ≠ 0.5), a pattern called “grain-strip cross pattern” is discovered, in the 0.43 <r <0.45.     

Multiway Dependence in Exponential Family Conditional Distributions

Conditionally specified statistical models are frequently constructed from one-parameter exponential family conditional distributions. One way to formulate such a model is to specify the dependence structure among random variables through the use of a Markov random field (MRF). A common assumption on the Gibbsian form of the MRF model is that dependence is expressed only through pairs of random variables, which we refer to as the “pairwise-only dependence” assumption. Based on this assumption, J. Besag (1974, J. Roy. Statist. Soc. Ser. B36, 192–225) formulated exponential family “auto-models” and showed the form that one-parameter exponential family conditional densities must take in such models. We extend these results by relaxing the pairwise-only dependence assumption, and we give a necessary form that one-parameter exponential family conditional densities must take under more general conditions of multiway dependence. Data on the spatial distribution of the European corn borer larvae are fitted using a model with Bernoulli conditional distributions and several dependence structures, including pairwise-only, three-way, and four-way dependencies.     

Polynomial identity rings as rings of functions

We generalize the usual relationship between irreducible Zariski closed subsets of the affine space, their defining ideals, coordinate rings, and function fields, to a non-commutative setting, where “varieties” carry a PGL_n-action, regular and rational “functions” on them are matrix-valued, “coordinate rings” are prime polynomial identity algebras, and “function fields” are central simple algebras of degree n. In particular, a prime polynomial identity algebra of degree n is finitely generated if and only if it arises as the “coordinate ring” of a “variety” in this setting. For n=1 our definitions and results reduce to those of classical affine algebraic geometry.     

Relative Stability for Strictly Stationary Sequences

For a nonnegative strictly stationary random sequence satisfying the “minimal” dependence condition necessary and sufficient conditions for the relative stability are found. As an application the well-known Khinchine stability result for i.i.d. random variables is proved for uniformly strong mixing sequences.     

Markov Uniqueness and Rademacher Theorem for Smooth Measures on an Infinite-Dimensional Space under Successful-Filtration Condition

For a smooth measure on an infinite-dimensional space, a “successful-filtration” condition is introduced and the Markov uniqueness and Rademacher theorem for measures satisfying this condition are proved. Some sufficient conditions, such as the well-known Hoegh-Krohn condition, are also considered. Examples demonstrating connections between these conditions and applications to convex measures are given.__________Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 2, pp. 170–186, February, 2005.     

Bicolored graph partitioning, or: gerrymandering at its worst

This study is motivated by an electoral application where we look into the following question: how much biased can the assignment of parliament seats be in a majority system under the effect of vicious gerrymandering when the two competing parties have the same electoral strength? To give a first theoretical answer to this question, we introduce a stylized combinatorial model, where the territory is represented by a rectangular grid graph, the vote outcome by a “balanced” red/blue node bicoloring and a district map by a connected partition of the grid whose components all have the same size. We constructively prove the existence in cycles and grid graphs of a balanced bicoloring and of two antagonist “partisan” district maps such that the discrepancy between their number of “red” (or “blue”) districts for that bicoloring is extremely large, in fact as large as allowed by color balance.     

Representing probability measures using probabilistic processes

In the Type-2 Theory of Effectivity, one considers representations of topological spaces in which infinite words are used as “names” for the elements they represent. Given such a representation, we show that probabilistic processes on infinite words, under which each successive symbol is determined by a finite probabilistic choice, generate Borel probability measures on the represented space. Conversely, for several well-behaved types of space, every Borel probability measure is represented by a corresponding probabilistic process. Accordingly, we consider probabilistic processes as providing “probabilistic names” for Borel probability measures. We show that integration is computable with respect to the induced representation of measures.     

Remarks on t-transformations of measures and convolutions

A family of transformations of probability measures is constructed, and used to define transformations of convolutions. The relations between moments and cumulants of a measure and its transformation are presented. For transformed classical and free convolutions the central limit measures and the Poisson type limit measures are computed. Families of non-commutative random variables are constructed, which are associated to these central limit measures. They provide examples of “position operators” which act on the Interacting Fock Spaces.     

Exploiting structure in quantified formulas

We study the computational problem “find the value of the quantified formula obtained by quantifying the variables in a sum of terms.” The “sum” can be based on any commutative monoid, the “quantifiers” need only satisfy two simple conditions, and the variables can have any finite domain. This problem is a generalization of the problem “given a sum-of-products of terms, find the value of the sum” studied in [R.E. Stearns and H.B. Hunt III, SIAM J. Comput. 25 (1996) 448–476]. A data structure called a “structure tree” is defined which displays information about “subproblems” that can be solved independently during the process of evaluating the formula. Some formulas have “good” structure trees which enable certain generic algorithms to evaluate the formulas in significantly less time than by brute force evaluation. By “generic algorithm,” we mean an algorithm constructed from uninterpreted function symbols, quantifier symbols, and monoid operations. The algebraic nature of the model facilitates a formal treatment of “local reductions” based on the “local replacement” of terms. Such local reductions “preserve formula structure” in the sense that structure trees with nice properties transform into structure trees with similar properties. These local reductions can also be used to transform hierarchical specified problems with useful structure into hierarchically specified problems having similar structure.     

Analysis in distribution of two randomized algorithms for finding the maximum in a broadcast communication model

The limit laws of three cost measures are derived of two algorithms for finding the maximum in a single-channel broadcast communication model. Both algorithms use coin flips and comparisons. Besides the ubiquitous normal limit law, the Dickman distribution also appears in a natural way. The method of proof proceeds along the line via the method of moments and the “asymptotic transfers,” which roughly bridges the asymptotics of the “conquering cost of the subproblems” and that of the total cost. Such a general approach has proved very fruitful for a number of problems in the analysis of recursive algorithms.     

On stability in terms of two measures of a hybrid system having partly invisible solutions

We consider three different dynamic systems. The first runs “smoothly” during a certain finite time interval, undergoes an abrupt change in the dynamics during the next (finite) time interval and is governed by the second system. The solution of this second system lies on a surface for a finite amount of time and becomes invisible. At the beginning of the third phase, the system is subjected to an impulse which causes the solution to leave the surface and we have the new hybrid impulsive system. In this paper, we employ two measures to find suitable conditions so that the new system again runs as smoothly as the first.     

Māl, enunciations, and the prehistory of Arabic algebra

Medieval Arabic algebra books intended for practical training generally have in common a first “book” which is divided into two sections: one on the methods of solving simplified equations and manipulating expressions, followed by one consisting of worked-out problems. By paying close attention to the wording of the problems in the books of al-Khwārizmī, Abū Kāmil, and Ibn Badr, we reveal the different ways the word māl was used. In the enunciation of a problem it is a common noun meaning “quantity,” while in the solution it is the proper noun naming the square of “thing” (shay '). We then look into the differences between the wording of enunciations and equations, which clarify certain problems solved without “thing,” and help explain the development of algebra before the time of al-Khwārizmī.     

The perception of randomness

Psychologists have studied people's intuitive notions of randomness by two kinds of tasks: judgment tasks (e.g., “is this series like a coin?” or “which of these series is most like a coin?”), and production tasks (e.g., “produce a series like a coin”). People's notion of randomness is biased in that they see clumps or streaks in truly random series and expect more alternation, or shorter runs, than are there. Similarly, they produce series with higher than expected alternation rates. Production tasks are subject to other biases as well, resulting from various functional limitations. The subjectively ideal random sequence obeys “local representativeness”; namely, in short segments of it, it represents both the relative frequencies (e.g., for a coin, 50%–50%) and the irregularity (avoidance of runs and other patterns). The extent to which this bias is a handicap in the real world is addressed.     

Random creation and dispersion of mass

Consider evolution of density of a mass or a population, geographically situated in a compact region of space, assuming random creation-annihilation and migration, or dispersion of mass, so the evolution is a random measure. When the creation-annihilation and dispersion are diffusions the situation is described formally by a stochastic partial differential equation; ignoring dispersion make approximations to the initial density by atomic measures and if the corresponding discrete random measures converge “in law” to a unique random measure call it a solution. To account for dispersion Trotter's product formula is applied to semiflows corresponding to dispersion and creation-annihilation. Existence of solutions has been a conjecture for several years despite a claim in ([2], J. Multivariate Anal. 5, 1–52). We show that solutions exist and that non-deterministic solutions are “smeared” continuous-state branching diffusions.     

The rate of convergence for the cyclic projections algorithm III: Regularity of convex sets

The cyclic projections algorithm is an important method for determining a point in the intersection of a finite number of closed convex sets in a Hilbert space. That is, for determining a solution to the “convex feasibility” problem. This is the third paper in a series on a study of the rate of convergence for the cyclic projections algorithm. In the first of these papers, we showed that the rate could be described in terms of the “angles” between the convex sets involved. In the second, we showed that these angles often had a more tractable formulation in terms of the “norm” of the product of the (nonlinear) metric projections onto related convex sets.In this paper, we show that the rate of convergence of the cyclic projections algorithm is also intimately related to the “linear regularity property” of Bauschke and Borwein, the “normal property” of Jameson (as well as Bakan, Deutsch, and Li’s generalization of Jameson’s normal property), the “strong conical hull intersection property” of Deutsch, Li, and Ward, and the rate of convergence of iterated parallel projections. Such properties have already been shown to be important in various other contexts as well.     

Twistless Tori Near Low-Order Resonances

In this paper, we investigate the behavior of the twist near low-order resonances of a periodic orbit or an equilibrium of a Hamiltonian system with two degrees of freedom. Namely, we analyze the case where the Hamiltonian has multiple eigenvalues (the Hamiltonian Hopf bifurcation) or a zero eigenvalue near the equilibrium and the case where the system has a periodic orbit whose multipliers are equal to 1 (the saddle-center bifurcation) or −1 (the period-doubling bifurcation). We show that the twist does not vanish at least in a small neighborhood of the period-doubling bifurcation. For the saddle-center bifurcation and the resonances of the equilibrium under consideration, we prove the existence of a “twistless” torus for sufficiently small values of the bifurcation parameter. An explicit dependence of the energy corresponding to the twistless torus on the bifurcation parameter is derived. Bibliography: 6 titles.__________Published in Zapiski Nauchnykh Seminarov POMI, Vol. 300, 2003, pp. 135–144.