Strong Law of Large Numbers and Central Limit Theorems for Functionals of Inhomogeneous Semi-Markov Processes

Limit theorems for functionals of classical (homogeneous) Markov renewal and semi-Markov processes have been known for a long time, since the pioneering work of Pyke Schaufele (Limit theorems for Markov renewal processes, Ann. Math. Statist., 35(4):1746–1764, 1964). Since then, these processes, as well as their time-inhomogeneous generalizations, have found many applications, for example, in finance and insurance. Unfortunately, no limit theorems have been obtained for functionals of inhomogeneous Markov renewal and semi-Markov processes as of today, to the best of the authors’ knowledge. In this article, we provide strong law of large numbers and central limit theorem results for such processes. In particular, we make an important connection of our results with the theory of ergodicity of inhomogeneous Markov chains. Finally, we provide an application to risk processes used in insurance by considering a inhomogeneous semi-Markov version of the well-known continuous-time Markov chain model, widely used in the literature.     

Abduction-induction (generalization) processes of elementary majors on figural patterns in algebra

The article deals with issues concerning the abductive-inductive reasoning of 42 preservice elementary majors on patterns that consist of figural and numerical cues. We discuss: ways in which the participants develop generalizations about classes of abstract objects; abductive processes they exhibit which support their induction leading to a generalization; ways they justify their generalizations in the abductive stage, and; the effects of figural and numerical cues in the manner they construct a plausible abductive generalization. Two types of abductions are explored, model-based and manipulative. A proposed abductive-inductive reasoning process for pattern sequences is presented and discussed in the concluding section.     

A thinning result for pure jump processes

A standard thinning procedure for point processes is extended to processes of pure jump type in which each jump is retained with probability p or deleted with probability 1 ? p, independently of everything else.Two theorems are proved, the first gives a sufficient condition for the existence of thinned pure jump processes, the second concerns the convergence of such processes to pure jump processes whose increments are generated by a Cox process. Some generalizations are discussed.     

Generalizations of the centroid with an application in stochastic geometry

The centroid of a subset of with positive volume is a well‐known characteristic. An interesting task is to generalize its definition to at least some sets of zero volume. In the presented paper we propose two possible ways how to do that. The first is based on the Hausdorff measure of an appropriate dimension. The second is given by the limit of centroids of ε‐neighbourhoods of the particular set when ε goes to 0. For both generalizations we discuss their existence and basic properties. Then we focus on sufficient conditions of existence of the second generalization and on conditions when both generalizations coincide. It turns out that they can be formulated with the help of the Minkowski content, rectifiability, and self‐similarity. Since the centroid is often used in stochastic geometry as a centre function for certain particle processes, we present properties that are needed for both generalizations to be valid centre functions. Finally, we also show their continuity on compact convex m‐sets with respect to the Hausdorff metric topology.     

Markov Processes Related with Dunkl Operators

Dunkl operators are parameterized differential-difference operators on ^Nthat are related to finite reflection groups. They can be regarded as a generalization of partial derivatives and play a major role in the study of Calogero–Moser–Sutherland-type quantum many-body systems. Dunkl operators lead to generalizations of various analytic structures, like the Laplace operator, the Fourier transform, Hermite polynomials, and the heat semigroup. In this paper we investigate some probabilistic aspects of this theory in a systematic way. For this, we introduce a concept of homogeneity of Markov processes on ^Nthat generalizes the classical notion of processes with independent, stationary increments to the Dunkl setting. This includes analogues of Brownian motion and Cauchy processes. The generalizations of Brownian motion have the càdlàg property and form, after symmetrization with respect to the underlying reflection groups, diffusions on the Weyl chambers. A major part of the paper is devoted to the concept of modified moments of probability measures on ^Nin the Dunkl setting. This leads to several results for homogeneous Markov processes (in our extended setting), including martingale characterizations and limit theorems. Furthermore, relations to generalized Hermite polynomials, Appell systems, and Ornstein–Uhlenbeck processes are discussed.     

Wishart processes

We propose some matrix generalizations of square Bessel processes and we indicate their first properties: hitting time of 0 of the smallest eigenvalue, additivity property, associated Martingales, distributions, which mainly extend the real-valued classical results. We explain why these processes are indecomposable and therefore differ from the real-valued ones. We conclude with some formulae concerning matrix quadratic functionals analogous to the Cameron Martin formula.     

Generalized potlatch and smoothing processes

Summary We consider simple generalizations of the potlatch and smoothing processes which were introduced in [8] and studied in [5]. These generalizations provide relatively simple examples of infinite interacting systems which exhibit phase transition. The original potlatch and smoothing processes do not exhibit phase transition. Our results show that for the generalized processes, phase transition does not usually occur in one or two dimensions, but usually does occur in higher dimensions. Upper and lower bounds for the relevant critical values are obtained. As one application of our results, we obtain the limiting behavior of the critical values for the linear contact process in d dimensions as d, thus answering a question we raised in [3]. This is done by comparing the contact process with an appropriate generalized smoothing process.Research supported in part by NSF Grant MCS 77-14881Research supported in part by NSF Grant MCS 77-02121     

A functional analytic approach to intersection bodies

We consider several generalizations of the concept of an intersection body and show their connections with the Fourier transform and embeddings in L _p -spaces. These connections lead to generalizations of the recent solution to the Busemann—Petty problem on sections of convex bodies. Submitted: October 1999, Revision: December 1999.     

Infinitely divisible distributions over locally compact non-archimedean fields

This paper is devoted to stochastic processes with values in finite-dimensional vector spaces over infinite, locally compact fields of zero and positive characteristics with nontrivial non-Archimedean norms. Infinitely divisible distributions are studied. Theorems about their characteristic functionals are proved. Particular cases are demonstrated as applications to non-Archimedean analogs of Gaussian and Poisson processes and their generalizations.     

Lagrange interpolation on extended generalized Jacobi nodes

We consider theL ^p -convergence of interpolatory processes for nonsmooth functions. Therefore we use generalizations of the well-known Marcinkiewicz-Zygmund inequality for trigonometric polynomials to the case of algebraic polynomials, extending a result of Y. Xu. Particularly, we obtain the order of convergence for certain Lagrange and quasi-Lagrange interpolatory processes on generalized Jacobi nodes. Our approach enables us also to discuss the influence of additional nodes near the endpoints ±1.     

Maximal branching processes with several types of particles

The paper introduces multitype maximal branching processes and their generalizations. Some examples of their explicit construction are given. The ergodic theorem is proved for the case of two types of particles.     

On the Boundary of the Union of Planar Convex Sets

We give two alternative proofs leading to different generalizations of the following theorem of [1]. Given n convex sets in the plane, such that the boundaries of each pair of sets cross at most twice, then the boundary of their union consists of at most 6n-12 arcs. (An arc is a connected piece of the boundary of one of the sets.) In the generalizations we allow pairs of boundaries to cross more than twice. Received May 5, 1997, and in revised form July 15, 1997.     

Cluster Algorithms for Spatial Point Processes with Symmetric and Shift Invariant Interactions

Abstract

In this article, Swendsen–Wang–Wolff algorithms are extended to simulate spatial point processes with symmetric and stationary interactions. Convergence of these algorithms is considered. Some further generalizations of the algorithms are discussed. The ideas presented in this article can also be useful in handling some large and complicated systems.     

Combinatorial Aspects of Flashcard Games

We study a family of discrete dynamical processes introduced by Novikoff, Kleinberg, and Strogatz that we call flashcard games. We prove a number of results on the evolution of these games, and in particular, we settle a conjecture of NKS on the frequency with which a given card appears. We introduce a number of generalizations and variations that we believe are of interest, and we provide a large number of open questions and problems.     

On the Maximum of Some Conditional and Integrated Gaussian Fields and their Statistical Applications

We evaluate upper bounds for the maximal distributions of some Gaussian random fields, which arise in the study of the asymptotic behavior of various two-dimensional empirical processes, with random index. Some of them are generalizations of well-known conditional Brownian fields, while the others are obtained by their integration. We present also some possible statistical applications of our results.     

Some contributions to the asymptotic theory of Bayes solutions

Summary This paper deals with the asymptotic theory of Bayes solutions in (i) Estimation (ii) Testing when hypothesis and alternative are separated at least by an indifference region, under the assumption that the observations are independent and indentically distributed. The estimation results which are partial generalizations of results of LeCam begin with a proof of the convergence of the normalized posterior density to the appropriate normal density in a strong sense. From this result we derive the asymptotic efficiency of Bayes estimates obtained from smooth loss functions and in particular of the posterior mean. The last two theorems of this section deal with asymptotic expansions for the posterior risk in such estimation problems. The section on testing contains a limit theorem for the n-th root of the posterior risk under weak conditions on the prior and the loss function. Finally we discuss generalizations and some open problems.Part of this research was done while P. J. Bickel was on leave at Imperial College, London. — This research was partially supported by National Science Foundation Grant GP-5059.This research was partially supported by National Science Foundation Grant GP-5705. Part of this research was done while J. A. Yahav was visiting the department of Statistics at Stanford University.     

Optimal non-isotropic gevrey exponents for sums of squares of vector fields

We prove sharp non-isotropic Gevrey hypoellipticiy for a class ofo partial differential operators that are sums of squares of real vector fields (and their generalizations) satisfying the Hormander bracket condition. These include the Baouendi-Goulaouic operator. Our results, which refine those of Chirst [8], are proved entirely by L ² methods and a careful study of brackets of vector fields. Applications to a recent conjecture of Traves are given.     

Bessel-type inequalities in Hilbert C *-modules

Regarding the generalizations of the Bessel inequality in Hilbert spaces which are due to Bombieri and Boas–Bellman, we obtain a version of the Bessel inequality and some generalizations of this inequality in the framework of Hilbert C *-modules.     

Asymmetric skew Bessel processes and their applications to finance

In this paper, we extend the Harrison and Shepp's construction of the skew Brownian motion (1981) and we obtain a diffusion similar to the two-dimensional Bessel process with speed and scale densities discontinuous at one point. Natural generalizations to multi-dimensional and fractional order Bessel processes are then discussed as well as invariance properties. We call this family of diffusions asymmetric skew Bessel processes in opposition to skew Bessel processes as defined in Barlow et al. [On Walsh's Brownian motions, Séminaire de Probabilitiés XXIII, Lecture Notes in Mathematics, vol. 1372, Springer, Berlin, New York, 1989, pp. 275–293]. We present factorizations involving (asymmetric skew) Bessel processes with random time. Finally, applications to the valuation of perpetuities and Asian options are proposed.     

Congruences on Menger algebras

We discuss some types of congruences on Menger algebras of rank n, which are generalizations of the principal left and right congruences on semigroups. We also study congruences admitting various types of cancellations and describe their relationship with strong subsets.