Pairwise sufficiency

Summary For an arbitrary experiment E we investigate the relation between its pairwise sufficient subalgebras and its sufficient sublattices in the M-space of E (in the sense of L. LeCam). By exhibiting an experiment without minimal pairwise sufficient subalgebra it is shown that this correspondence is in general not bijective. In view of this we introduce the rather large class of majorized experiments. They have a minimal pairwise sufficient subalgebra which can be described explicitely.As a natural subclass of the majorized experiments appear the coherent experiments that are distinguished by the coincidence of sufficiency and pairwise sufficiency. It is shown that the coherent experiments are characterized by the fact that they admit a majorizing measure which is localizable. As a consequence we obtain that the class of coherent experiments coincides with classes of experiments previously introduced by T.S. Pitcher, D. Mußmann, M. Hasegawa and M.D. Perlman.     

A NOTE ON A PAPER BY NG PENG-NUNG AND LEE PENG-YEE

Abstract

In the paper “Convergence in normed Köthe spaces” (J. Singapore National Academy of Science, 4, 146–148 (1975) M.R. 52 # 11568) Ng Peng-Nung and Lee Peng-Yee obtained a convergence result in the general setting of Banach funcation spaces providing conditions in order that pointwise and weak convergence imply norm convergence. They claim this result to be a generalization of a corresponding well known result in the Lebesgue space L₁ (X, u). To substantiate their claim it is necessary to show that the class of Banach function spaces for which their theorem holds is larger than the class of L₁-spaces. This, we shall show, is unfortunately not the case.     

Dominated Pesin theory: convex sum of hyperbolic measures

In the uniformly hyperbolic setting it is well known that the set of all measures supported on periodic orbits is dense in the convex space of all invariant measures. In this paper we consider the converse question, in the non-uniformly hyperbolic setting: assuming that some ergodic measure converges to a convex combination of hyperbolic ergodic measures, what can we deduce about the initial measures?To every hyperbolic measure μ whose stable/unstable Oseledets splitting is dominated we associate canonically a unique class H(μ) of periodic orbits for the homoclinic relation, called its intersection class. In a dominated setting, we prove that a measure for which almost every measure in its ergodic decomposition is hyperbolic with the same index, such as the dominated splitting, is accumulated by ergodic measures if, and only if, almost all such ergodic measures have a common intersection class.We provide examples which indicate the importance of the domination assumption.     

Rigid sets and nonexpansive mappings

We introduce a new class of normed spaces (not necessarily finite dimensional), which contains the finite dimensional normed spaces with polyhedral norm. We study the properties of rigid sets of the spaces of this class and we apply the results to limit sets of the sequences of iterates of nonexpansive maps.

     

On The Spaces of Linear Operators Acting Between Asymmetric Cone Normed Spaces

An asymmetric norm is a positive sublinear functional p on a real vector space X satisfying \(x=\theta _X\) whenever \(p(x)=p(-x)=0\). Since the space of all lower semi-continuous linear functionals of an asymmetric normed space is not a linear space, the theory is different in the asymmetric case. The main purpose of this study is to define bounded and continuous linear operators acting between asymmetric cone normed spaces. After examining the differences with symmetric case, we give some results related to Baire’s characterization of completeness in asymmetric cone normed spaces.     

A set scalarization function based on the oriented distance and relations with other set scalarizations

ABSTRACT

The aim of this paper is, in the setting of normed spaces with a cone K non necessarily solid, to study new relations among set scalarization functions that are extensions of the oriented distance of Hiriart-Urruty. Moreover, we deal with a set scalarization function of sup-inf type, we investigate its relation to the cone-properness and cone-boundedness and it is related to other set scalarizations existing in the literature. In particular, with the norm induced by the Minkowski's functional, we obtain relations with a set scalarization which is an extension of the so called Gerstewitz's scalarization function.     

Some remarks on functions with values in probabilistic normed spaces

In this paper we consider an enlargement of the notion of the probabilistic normed space. For this new class of probabilistic normed spaces we give some topological properties. By using properties of the probabilistic norm we prove some differential and integral properties of functions with values into probabilistic normed spaces. As special cases, results for deterministic and random functions can be obtained.      

On a characterization of monotone likelihood ratio experiments

Summary Pfanzagl (1962,Zeit. Wahrscheinlichkeitsth.,1, 109–115) showed that a dominated family of probability measures has monotone likelihood ratios with respect to some real valued statistic if there exists a set of tests which has certain nice properties. A similar characterization was given by Dettweiler (1978,Metrika,25, 247–254), who did not assume domination. However, Pfanzagl's result is not a special case of the one proved by Dettweiler. We present a theorem which comprises the results of both authors. Our proof shows that not all conditions introduced by them are needed. Furthermore, we investigate the question concerning the generality we get if we do not assume domination.     

On ideal convergence in probabilistic normed spaces

An interesting generalization of statistical convergence is I-convergence which was introduced by P.Kostyrko et al [KOSTYRKO,P.—ŠALáT,T.—WILCZYŃSKI,W.: I-Convergence, Real Anal. Exchange 26 (2000–2001), 669–686]. In this paper, we define and study the concept of I-convergence, I*-convergence, I-limit points and I-cluster points in probabilistic normed space. We discuss the relationship between I-convergence and I*-convergence, i.e. we show that I*-convergence implies the I-convergence in probabilistic normed space. Furthermore, we have also demonstrated through an example that, in general, I-convergence does not imply I*-convergence in probabilistic normed space.     

Sublinear functionals and conical measures

The paper is devoted to the concept of conical measures which is central for the Choquet theory of integral representation in its final version. The conical measures need not be continuous under monotone pointwise convergence of sequences on the lattice subspace of functions which form their domain. We prove that they indeed become continuous (even in the nonsequential sense) when one restricts that domain to an obvious subcone. This result is in accord with the recent representation theory in measure and integration developed by the author. We also prove that one can pass from the subcone in question to a certain natural extended cone.     

On complete convergence in mean for double sums of independent random elements in Banach spaces

Conditions are provided under which a normed double sum of independent random elements in a real separable Rademacher type p Banach space converges completely to 0 in mean of order p. These conditions for the complete convergence in mean of order p are shown to provide an exact characterization of Rademacher type p Banach spaces. In case the Banach space is not of Rademacher type p, it is proved that the complete convergence in mean of order p of a normed double sum implies a strong law of large numbers.     

Minimal convex majorants of functions and Demyanov–Rubinov exhaustive super(sub)differentials

ABSTRACT

The primary goal of the paper is to establish characteristic properties of (extended) real-valued functions defined on normed vector spaces that admit the representation as the lower envelope (the pointwise infimum) of their minimal (with respect of the pointwise ordering) convex majorants. The results presented in the paper generalize and extend the well-known Demyanov-Rubinov characterization of upper semicontinuous positively homogeneous functions as the lower envelope of exhaustive families of continuous sublinear functions to larger classes of (not necessarily positively homogeneous) functions defined on arbitrary normed spaces. As applications of the above results, we introduce, for nonsmooth functions, a new notion of the Demyanov-Rubinov exhaustive subdifferential at a given point, and show that it generalizes a number of known notions of subdifferentiability, in particular, the Fenchel-Moreau subdifferential of convex functions, the Dini-Hadamard (directional) subdifferential of directionally differentiable functions, and the Φ-subdifferential in the sense of the abstract convexity theory. Some applications of Demyanov-Rubinov exhaustive subdifferentials to extremal problems are considered.     

On the definition of a probabilistic normed space

Summary In this paper we give a new definition of a probabilistic normed space. This definition, which is based on a characterization of normed spaces by means of a betweenness relation, includes the earlier definition of A. N. erstnev as a special case and leads naturally to the definition of the principal class of probabilistic normed spaces, the Menger spaces.     

Constraint qualifications for nonsmooth programming

ABSTRACT

By using an implicit function theorem and a result of error bound, we provide new constraint qualifications ensuring the Karush–Kuhn–Tuker necessary optimality conditions for both smooth and nonsmooth optimization problems in normed spaces or Banach spaces.     

TWO POROSITY RESULTS IN MONOTONIC ANALYSIS

ABSTRACT

In this work we consider spaces of increasing functions defined on a subset of an ordered normed space. We equip each of these spaces with a natural metric and show that the complement of the subset of all strictly increasing functions is σ-porous. We also discuss some properties of normal sets and strictly normal sets.     

Simultaneous metric projection onto closed sets

We examine simultaneous metric projection by closed sets in a class of ordered normed spaces. First, we study simultaneous metric projection onto downward and upward sets and separation properties of these sets. The results obtained are used for examination of simultaneous metric projection by arbitrary closed sets, and we examine the minimization of the distance from a bounded set to an arbitrary closed set in a class of ordered normed spaces.     

On square roots of isometries

Abstract

In various normed spaces we answer the question of when a given isometry is a square of some isometry. In particular, we consider (real and complex) matrix spaces equipped with unitarily invariant norms and unitary congruence invariant norms, as well as some infinite dimensional spaces illustrating the difference between finite and infinite dimensions.     

Local <Emphasis Type="Italic">Tb</Emphasis> theorem with <Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript> testing conditions and general measures: square functions

Local Tb theorems with L ^p type testing conditions, which are not scale invariant, have been studied widely in the case of the Lebesgue measure. In the non-homogeneous world local Tb theorems have only been proved assuming scale invariant (L ^∞ or BMO) testing conditions. In this paper, for the first time, we overcome these obstacles in the non-homogeneous world, and prove a nonhomogeneous local Tb theorem with L ² type testing conditions. This paper is in the setting of the vertical and conical square functions defined using general measures and kernels. On the technique side, we demonstrate a trick of inserting Calderón–Zygmund stopping data of a fixed function into the construction of the twisted martingale difference operators. This built-in control of averages is an alternative to Carleson embedding.     

A Real Number Structure that is Effectively Categorical

On countable structures computability is usually introduced via numberings. For uncountable structures whose cardinality does not exceed the cardinality of the continuum the same can be done via representations. Which representations are appropriate for doing real number computations? We show that with respect to computable equivalence there is one and only one equivalence class of representations of the real numbers which make the basic operations and the infinitary normed limit operator computable. This characterizes the real numbers in terms of the theory of effective algebras or computable structures, and is reflected by observations made in real number computer arithmetic. Demanding computability of the normed limit operator turns out to be essential: the basic operations without the normed limit operator can be made computable by more than one class of representations. We also give further evidence for the well-known non-appropriateness of the representation to some base b by proving that strictly less functions are computable with respect to these representations than with respect to a standard representation of the real numbers. Furthermore we consider basic constructions of representations and the countable substructure consisting of the computable elements of a represented, possibly uncountable structure. For countable structures we compare effectivity with respect to a numbering and effectivity with respect to a representation. Special attention is paid to the countable structure of the computable real numbers.