Refined discrete regular variation and its applications

We introduce a new class of the so‐called regularly varying sequences with respect to τ and state its properties. This class, on one hand, generalizes regularly varying sequences. On the other hand, it refines them and makes it possible to do a more sophisticated analysis in applications. We show a close connection with regular variation on time scales; thanks to this relation, we can use the existing theory on time scales to develop discrete regular variation with respect to τ. We reveal also a connection with generalized regularly varying functions. As an application, we study asymptotic behavior of solutions to linear difference equations; we obtain generalization and extension of known results. The theory also yields, in some way, a new view on the tests for convergence and divergence of series; we establish the statement that generalizes Raabe test and Bertrand test.     

A unified Floquet theory for discrete, continuous, and hybrid periodic linear systems

In this paper, we study periodic linear systems on periodic time scales which include not only discrete and continuous dynamical systems but also systems with a mixture of discrete and continuous parts (e.g. hybrid dynamical systems). We develop a comprehensive Floquet theory including Lyapunov transformations and their various stability preserving properties, a unified Floquet theorem which establishes a canonical Floquet decomposition on time scales in terms of the generalized exponential function, and use these results to study homogeneous as well as nonhomogeneous periodic problems. Furthermore, we explore the connection between Floquet multipliers and Floquet exponents via monodromy operators in this general setting and establish a spectral mapping theorem on time scales. Finally, we show this unified Floquet theory has the desirable property that stability characteristics of the original system can be determined via placement of an associated (but time varying) system?s poles in the complex plane. We include several examples to show the utility of this theory.     

An asymptotic analysis of positive solutions of generalized Thomas-Fermi differential equations — The sub-half-linear case

The existence and the asymptotics behavior for the large value of the variable of the positive solutions of generalized Thomas-Fermi equation presented in this article are proved. It is assumed that coefficient q(t) belongs to the class of regularly varying functions in the sense of Karamata. Properties of these functions and the Schauder-Tychonoff fixed point theorem are the main tools for the proofs.     

Extremal behavior of regularly varying stochastic processes

We study a formulation of regular variation for multivariate stochastic processes on the unit interval with sample paths that are almost surely right-continuous with left limits and we provide necessary and sufficient conditions for such stochastic processes to be regularly varying. A version of the Continuous Mapping Theorem is proved that enables the derivation of the tail behavior of rather general mappings of the regularly varying stochastic process. For a wide class of Markov processes with increments satisfying a condition of weak dependence in the tails we obtain simplified sufficient conditions for regular variation. For such processes we show that the possible regular variation limit measures concentrate on step functions with one step, from which we conclude that the extremal behavior of such processes is due to one big jump or an extreme starting point. By combining this result with the Continuous Mapping Theorem, we are able to give explicit results on the tail behavior of various vectors of functionals acting on such processes. Finally, using the Continuous Mapping Theorem we derive the tail behavior of filtered regularly varying Lévy processes.     

Extremal solutions to a system of n nonlinear differential equations and regularly varying functions

The strongly increasing and strongly decreasing solutions to a system of n nonlinear first order equations are here studied, under the assumption that both the coefficients and the nonlinearities are regularly varying functions. We establish conditions under which such solutions exist and are (all) regularly varying functions, we derive their index of regular variation and establish asymptotic representations. Several applications of the main results are given, involving n‐th order nonlinear differential equations, equations with a generalized ?‐Laplacian, and nonlinear partial differential systems.     

The concept of spectral dichotomy for linear difference equations II

This paper is a continuation of a previous one (J. Math. Anal. Appl. 185 (1994), 275–287) in which the concept of spectral dichotomy has been introduced. This new notion of dichotomy has proved to be useful since it allows to apply the well known theory of linear operators to study dynamic properties of nonautonomous linear difference equations. In the present paper we extend our result on the equivalence of the spectral dichotomy and the well known exponential dichotomy to the class of linear differenc equations whose right-hand sides are not necessarily invertible. We furthermore investigate equations on the set of positive integers for which we establish necessary and sufficient conditions for exponential and unifrom stability.     

Admissibility and minimaxity of generalized Bayes estimators for spherically symmetric family

We give a sufficient condition for admissibility of generalized Bayes estimators of the location vector of spherically symmetric distribution under squared error loss. Compared to the known results for the multivariate normal case, our sufficient condition is very tight and is close to being a necessary condition. In particular, we establish the admissibility of generalized Bayes estimators with respect to the harmonic prior and priors with slightly heavier tail than the harmonic prior. We use the theory of regularly varying functions to construct a sequence of smooth proper priors approaching an improper prior fast enough for establishing the admissibility. We also discuss conditions of minimaxity of the generalized Bayes estimator with respect to the harmonic prior.     

Topological regular variation: III. Regular variation

This paper extends the topological theory of regular variation of the slowly varying case of Bingham and Ostaszewski (2010) [5] to the regularly varying functions between metric groups, viewed as normed groups (see also Bingham and Ostaszewski (2010) [6]). This employs the language of topological dynamics, especially flows and cocycles. In particular we show that regularly varying functions obey the chain rule and in the non-commutative context we characterize pairs of regularly varying functions whose product is regularly varying. The latter requires the use of a ‘differential modulus’ akin to the modulus of Haar integration.     

Basic qualitative and quantitative results for solutions to nonlinear,dynamic equations on time scales with an application to economic modelling

This article investigates both basic qualitative and basic quantitative properties of solutions to first- and higher-order dynamic equations on time scales and thus provides a foundation and framework for future advanced nonlinear studies in the field. Particular focus lies in the: existence; uniqueness; dependency; approximation; and explicit representation, of solutions to nonlinear initial value problems. The main tools used are from modern areas of nonlinear analysis, including: the fixed-point theorems of Banach and Schäfer; the method of successive approximations; a novel definition of measuring distance in metric spaces and normed spaces; and a “separation” of variables technique is introduced to the general time scale setting.     

Periodic solutions for dynamic equations on time scales

Massera type criteria are established for the existence of periodic solutions of linear and nonlinear dynamic equations on time scales. Some interesting properties of the exponential function on time scales are presented. Furthermore, a sufficient condition guaranteeing the boundedness of the solutions of the equation is presented.     

A class of the first order impulsive dynamic equations on time scales

In this paper, the first order impulsive dynamic equations on time scales is considered. The existence and uniqueness of solution for a class of nonlinear impulsive dynamic equations on time scales is proved.     

Operator Tail Dependence of Copulas

A notion of tail dependence based on operator regular variation is introduced for copulas, and the standard tail dependence used in the copula literature is included as a special case. The non-standard tail dependence with marginal power scaling functions having possibly distinct tail indexes is investigated in detail. We show that the copulas with operator tail dependence, incorporated with regularly varying univariate margins, give rise to a rich class of the non-standard multivariate regularly varying distributions. We also show that under some mild conditions, the copula of a non-standard multivariate regularly varying distribution has the standard tail dependence of order 1. Some illustrative examples are given.     

On functions behaving like additive functions

Let 0 < 1. In the paper we consider the following inequality: |f(x + y) – f(x) – f(y)| min{|f(x + y)|, |f(x) + f(y)|}, wheref: R R. Solutions and continuous solutions of this inequality are investigated. They have similar properties as additive functions, e.g. if the solution is bounded above (below) on a set of positive inner Lebesgue measure then it is continuous. Some sufficient condition for this inequality is also given.Dedicated to Professor Otto Haupt with best wishes on his 100th birthday     

Scaling properties of Hausdorff and packing measures

Let . Let be a continuous increasing function defined on , for which and is a decreasing function of t. Let be a norm on , and let , , denote the corresponding metric, and Hausdorff and packing measures, respectively. We characterize those functions such that the corresponding Hausdorff or packing measure scales with exponent by showing it must be of the form , where L is slowly varying. We also show that for continuous increasing functions and defined on , for which , is either trivially true or false: we show that if , then for a constant c, where is the Lebesgue measure on . Received June 17, 2000 / Accepted September 6, 2000 / Published online March 12, 2001     

Topological regular variation: I. Slow variation

Motivated by the Category Embedding Theorem, as applied to convergent automorphisms (Bingham and Ostaszewski (in press) [11]), we unify and extend the multivariate regular variation literature by a reformulation in the language of topological dynamics. Here the natural setting are metric groups, seen as normed groups (mimicking normed vector spaces). We briefly study their properties as a preliminary to establishing that the Uniform Convergence Theorem (UCT) for Baire, group-valued slowly-varying functions has two natural metric generalizations linked by the natural duality between a homogenous space and its group of homeomorphisms. Each is derivable from the other by duality. One of these explicitly extends the (topological) group version of UCT due to Bajšanski and Karamata (1969) [4] from groups to flows on a group. A multiplicative representation of the flow derived in Ostaszewski (2010) [45] demonstrates equivalence of the flow with the earlier group formulation. In companion papers we extend the theory to regularly varying functions: we establish the calculus of regular variation in Bingham and Ostaszewski (2010) [13] and we extend to locally compact, σ-compact groups the fundamental theorems on characterization and representation (Bingham and Ostaszewski (2010) [14]). In Bingham and Ostaszewski (2009) [15], working with topological flows on homogeneous spaces, we identify an index of regular variation, which in a normed-vector space context may be specified using the Riesz representation theorem, and in a locally compact group setting may be connected with Haar measure.     

Multivariate linear recursions with Markov-dependent coefficients

We study a linear recursion with random Markov-dependent coefficients. In a “regular variation in, regular variation out” setup we show that its stationary solution has a multivariate regularly varying distribution. This extends results previously established for i.i.d. coefficients.     

Numerical treatment of oscillatory functional differential equations

In this paper we are concerned with oscillatory functional differential equations (that is, those equations where all the solutions oscillate) under a numerical approximation. Our interest is in the preservation of qualitative properties of solutions under a numerical discretisation. We give conditions under which an equation is oscillatory, and consider whether the discrete schemes derived using linear ?-methods will also be oscillatory. We conclude with some general theory.     

On the stability of the translation equation and dynamical systems

In this paper we prove the stability of the functional equation F(s,F(t,x))=F(s+t,x) in the class of functions F:R×I→I, which are continuous with respect to each variable, and where I⊂R is a real interval. We also discuss the stability in the sense of Hyers-Ulam of dynamical systems on I. We show some properties of δ-approximate solutions of the translation equation on a real interval.     

On the tail behavior of a class of multivariate conditionally heteroskedastic processes

Conditions for geometric ergodicity of multivariate autoregressive conditional heteroskedasticity (ARCH) processes, with the so-called BEKK (Baba, Engle, Kraft, and Kroner) parametrization, are considered. We show for a class of BEKK-ARCH processes that the invariant distribution is regularly varying. In order to account for the possibility of different tail indices of the marginals, we consider the notion of vector scaling regular variation (VSRV), closely related to non-standard regular variation. The characterization of the tail behavior of the processes is used for deriving the asymptotic properties of the sample covariance matrices.     

A metric approach to asymptotic analysis

We introduce a notion of “firm” (or uniform) asymptotic cone to an unbounded subset of a normed space. We relate this notion to a concept of “firm” asymptotic function. We use these notions to study boundedness properties which can be applied to continuity questions for some operations on sets and functions. Such questions arise in stability analysis of Hamilton-Jacobi equations. We present some other applications such as an extension of a theorem of Dieudonné and existence results in optimization and fixed point theory.