Convolution Operators with Fractional Measures Associated to Holomorphic Functions

Let be an open set in the complex plane and let be a holomorphic function on . Let K be a compact subset of with nonempty interior such that 0 K. Let be the Borel measure of R ⁴ C ² given by(E = _K E(z, (z))|z|^–2 d(z)where 0 < 2 and d(x ₁ + ix ₂) = dx ₁ dx ₂ denotes the Lebesgue measure on C. Let T be the convolution operator T f = * f. In this paper we characterize the type set E associated to T .     

On harmonic renewal measures

Summary Let be a probability and the corresponding harmonic renewal measure. Complementing earlier results where is concentrated on a halfline we investigate the behaviour ofv _h ([x, x + 1]) and the harmonic renewal functionG(x) =v _h((–,x])asx ifm ₁=x(dx)>0. We also consider the casem ₁=0.     

Jensen inequalities for functions with higher monotonicities

Summary We investigate generalizations of the classical Jensen and Chebyshev inequalities. On one hand, we restrict the class of functions and on the other we enlarge the class of measures which are allowed. As an example, consider the inequality (J)(f(x) d) A (f(x) d, d d = 1. Iff is an arbitrary nonnegativeL _x function, this holds if 0, is convex andA = 1. Iff is monotone the measure need not be positive for (J) to hold for all convex withA = 1. If has higher monotonicity, e.g., is also convex, then we get a version of (J) withA < 1 and measures that need not be positive.     

On the Approximation of Certain Mass Distributions Appearing in Distance Geometry

Let X be a compact subset of the n-dimensional Euclidean space R ⁿ . A theorem of G. Björck implies the existence of a unique probability measure ₀ which maximizes the value _{X X} d ₂(x, y) d(x) d(y), where ranges over all probability measures on X and d ₂ denotes the Euclidean distance on R ⁿ . In this paper we introduce and investigate an algorithm which is easy to describe and which inductively constructs a sequence = x ₁, x ₂,... in X such that is uniformly distributed with respect to ₀. Geometrical and topological interpretations and applications, and concrete numerical examples are given.     

Maximum Monoreflections and Essential Extensions

An extension operator c in a category is an assignment, to each object A a monomorphism c _A : AcA. Seeking to approximate such a c by a functor, in our earlier paper Maximum monoreflections, we showed that with some hypotheses on the category, and on c, there is a monoreflection (c) maximum beneath c. Thus, in a suitable category of rings, using the complete ring of quotients operator Q, each object A has a maximum functorial ring of quotients (Q)A. But the proof gave no hint of how to calculate the general (c)A's, nor the particular (Q)A's. In the present paper, we give an explicit formula (and separate proof of existence) for the (c)A's, under more complicated hypotheses on the category and assuming the c _A 's are essential monomorphisms. We discuss briefly how the formula proves adequate to calculate the (Q)A's in Archimedean f-rings, and some related and necessary constructs in Archimedean l-groups.     

Limiting behavior of weighted central paths in linear programming

We study the limiting behavior of the weighted central paths{(x(), s())} > 0 in linear programming at both = 0 and = . We establish the existence of a partition (B ,N ) of the index set { 1, ,n } such thatx _i() ands _j () as fori B , andj N , andx _N (),s _B () converge to weighted analytic centers of certain polytopes. For allk 1, we show that thekth order derivativesx ^(k) () ands ^(k) () converge when 0 and . Consequently, the derivatives of each order are bounded in the interval (0, ). We calculate the limiting derivatives explicitly, and establish the surprising result that all higher order derivatives (k 2) converge to zero when .     

Weakly compactly generated Banach spaces and the strong law of large numbers

IfB is a weakly compactly generated Banach space andf: (S,S, ) satisfies the strong law of large numbers, thenf=f ₁+f ₂, wheref ₁ is Bochner -integrable andf ₂ is Pettis -integrable with Pettis norm 0. The decomposition is unique.     

Operator stable laws: Series representations and domains of normal attraction

LetX,X ₁,X ₂,... be i.i.d. random vectors in ^d. The limit laws that can arise by suitable affine normalizations of the partial sums,S _n=X ₁+...+X _n, are calledoperator-stable laws. These laws are a natural extension to ^d of the stable laws on. Thegeneralized domain of attraction of [GDOA()] is comprised of all random vectorsX whose partial sums can be affinely normalized to converge to . If the linear part of the affine transformation is restricted to take the formn ^–B for some exponent operatorB naturally associated to thenX is in thegeneralized domain of normal attraction of [GDONA()]. This paper extends the theory of operator-stable laws and their domains of attraction and normal attraction.     

Sur certaines algèbres associées à une mesure de Guelfand

Let be a Guelfand measure (cf. [A, B]) on a locally compact groupG DenoteL ₁ (G)=*L ₁(G)* the commutative Banach algebra associated to . We show thatL ₁ (G) is semi-simple and give a characterization of the closed ideals ofL ₁ (G). Using the -spherical Fourier transform, we characterize all linear bounded operators inL ₁ (G) which are invariants by -translations (i.e. such that ¹(( _x f) )=( _x ((f)) for eachxG andfL ₁ (G); where _x f(y)=f(xy); x,y G). WhenG is compact, we study the algebraL ₁ (G) and obtain results analogous to ones obtained for the commutative case: we show thatL ₁ (G) is regular, all closed sets of its Guelfand spectrum are sets of synthesis and establish theorems of harmonic synthesis for functions inL _p (G) (p=1,2 or +).

     

Fubini theorem for anticipating stochastic integrals in Hilbert space

Let (X,l,) be a measure space, letW be a cylindrical Hilbert-Wiener process, and let be an anticipating integrable process-valued function onX. We prove, under natural assumptions on, that there exists a measurable version Y_x,x X, of the anticipating integral of(x) such that the integral _x Y_x(dx) is a version of the anticipating integral of _X (x)(dx). We apply this anticipating Fubini theorem to study solutions of a class of stochastic evolution equations in Hilbert space.     

Effilement minimal en une singularité isolée de l'équation de Schrödinger et application au principe de Picard

In this paper, the generalized Schrödinger equation (–)u=0 on the punctured unit disk of ² is investigated. If is rotation free and satisfies the Picard principle at the origin, it is shown that if a setE is minimal thin relatively to an extremal harmonic functionh with zero boundary values at {|x|=1}, there exists a sequence (r _n ) converging to zero such that B(O,r _n ) C E. Lete be the -unit. It is proved that if a measure satisfies _\E e h d<, for a minimal thin, relatively toh , setE then the Picard principle is valid for the measure + .

     

Measures not charging semipolars and equations of schrödinger type

SupposeX is a Borel right process andm is a -finite excessive measure forX. Given a positive measure not chargingm-semipolars we associate an exact multiplicative functionalM(). No finiteness assumptions are made on . Given two such measures and ,M()=M() if and only if and agree on all finely open measurable sets. The equation (q–L)u+u=f whereL is the generator of (a subprocess of)X may be solved for appropriatef by means of the Feynman-Kac formula based onM(). Both uniqueness and existence are considered.Supported in part by NSF Grant DMS 92-24990.     

Metrics with finite sets of primitive extensions

In this paper, we give a complete characterization for the class of rational finite metrics with the property that the set () of primitive extensions of is finite. Here, for a metric on a setT, a positive extensionm of to a setV T is calledprimitive if none of the convex combinations of other extensions of toV is less than or equal tom. Our main theorem asserts that the following the properties are equivalent: (i) () is finite; (ii) Up to an integer factor, is a submetric of the path metric d ^H of a graphH with |(d ^H )=1; (iii) A certain bipartite graph associated with contains neither isometrick-cycles withk6 nor induced subgraphsK _3,3 ^– . We then show that () is finite if and only if the dimension of the tight span of is at most two. We also present other results, discuss applications to multicommodity flows, and raise open problems.This research was supported by grant 97-01-00115 from the Russian Foundation of Basic Research and a grant from the Sonderforschungsbereich 343, Bielefeld Universität, Bielefeld, Germany.     

Weak Mixing of Random Walks on Groups

Let G be a locally compact -compact group with right Haar measure m and a regular probability measure on G. We say that is weakly mixing if for all gL (G) and all fL ¹(G) with fdm=0 we have n ^{–1 n} _k=1| ^k *f,g|0. We show that is weakly mixing if and only if is ergodic and strictly aperiodic. To prove this we use and prove some results about unimodular eigenvalues for general Markov operators.     

The Product of Independent Random Variables with Regularly Varying Tails

A distribution is said to have regularly varying tail with index – (0) if lim _x(kx,)/(x,)=k ^– for each k>0. Let X and Y be independent positive random variables with distributions and , respecitvely. The distribution of product XY is called Mellin–Stieltjes convolution (MS convolution) of and . It is known that D() (the class of distributions on (0,) that have regularly varying tails with index –) is closed under MS convolution. This paper deals with decomposition problem of distributions in D() related to MS convolution. A representation of a regularly varying function F of the following form is investigated: F(x)= _k=0 ^n–1 b _k f(a ^k x), where f is a measurable function and a and b _k (k=1,...,n–1) are real constants. A criterion is given for these constants in order that f be regularly varying. This criterion is applicable to show that there exist two distributions and such that neither nor belongs to D() (>0) and their MS convolution belongs to D().     

Edgeworth expansions of suitably normalized sample mean statistics for atomic Markov chains

This paper is devoted to the problem of estimating functionals of type (f)=fd from observations drawn from a positive recurrent atomic Markov chain with stationary distribution . The properties of different estimators are studied. Beyond an accurate estimation of their bias, the estimation of their asymptotic variance is considered. We also show that the results of Malinovskii(1987) on the validity of the formal Edgeworth expansion for sample mean statistics of type T_n = n^–1n_i=1f(X_i) extend to their studentized versions, normalized by the asymptotic variance estimates we consider.

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Resume. Cet article est consacré au problème de lestimation dune fonctionnelle linéaire (f)=fd à partir de lobservation dune chaîne de Markov récurrente positive possédant un atome accessible et de distribution stationnaire . Les propriétés de plusieurs estimateurs sont étudiées. Au delà dune estimation précise de leurs biais respectifs, nous nous intéressons également à lestimation de la variance asymptotique de ces estimateurs. Nous montrons aussi que les résultats de Malinovskii (1987) concernant le développement dEdgeworth de lestimateur T_n = n^–1n_i=1f(X_i) sétendent à la version studentisée, lorsquil est normalisé par lestimateur de la variance asymptotique que nous proposons.

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Revised version: 3 March 2004     

The decomposition theorem for functions satisfying the law of large numbers

LetB be a Banach space with the Radon-Nikodym property and (S, , ) a probability space. Then anf: SB satisfies the strong law of large numbers if and only if there exists a Bochner integrable functionf ₁ and a Pettis integrable functionf ₂,f ₂f ₂=0 in the Glivenko-Cantelli norm, such thatf=f ₁+f ₂. The composition is unique.     

Representations of integers as sums of powers with increasing exponents

Let(n) be the least integer such thatn may be represented in the formn=x ₁ ² +x ₂ ³ +...+x _(n) ⁽ⁿ⁾⁺¹ wherex ₁,x ₂, ...,x _(n) are natural numbers. We computed(n) forn 250 000 and found that(n) 5 for all thesen exceptn=56, 160 for which(n)=6. Also(n) 4 for 41542<n<=250 000.     

A graded scale of parametric families of distributions,and parameter estimates based on the sample mean

Summary Denote by _k a class of familiesP={P} of distributions on the line R¹ depending on a general scalar parameter , being an interval of R¹, and such that the moments µ₁()=xdP ,...,µ_2k ()=x ^2k dP are finite, ₁ (), ..., _k (), _k+1 () ..., _k () exist and are continuous, with ₁ () 0, and _j +1 ()= ₁ () _j () +[₂() -₁()²] _j ()/ ₁ (), J=2, ..., k. Let ₁=¯x=x ₁ + ... +x _n/n, ₂=x ₁ ² + ... +x _n ²/n, ..., _k =(x ₁ ^k + ... +x _n ^k/n denote the sample moments constructed for a sample x₁, ..., x_n from a population with distribution Pg. We prove that the estimator of the parameter by the method of moments determined from the equation ₁= ₁() and depending on the observations x₁, ..., x_n only via the sample mean ¯x is asymptotically admissible (and optimal) in the class _k of the estimators determined by the estimator equations of the form ₀ () + ₁ () ₁ + ... + _k () _k =0 if and only ifP _k .The asymptotic admissibility (respectively, optimality) means that the variance of the limit, as n (normal) distribution of an estimator normalized in a standard way is less than the same characteristic for any estimator in the class under consideration for at least one 9 (respectively, for every ).The scales arise of classes _{1 2}... of parametric families and of classes _{1 2} ... of estimators related so that the asymptotic admissibility of an estimator by the method of moments in the class _k is equivalent to the membership of the familyP in the class _k .The intersection consists only of the families of distributions with densities of the form h(x) exp {C₀() + C₁() x } when for the latter the problem of moments is definite, that is, there is no other family with the same moments ₁ (), ₂ (), ...Such scales in the problem of estimating the location parameter were predicted by Linnik about 20 years ago and were constructed by the author in [1] (see also [2, 3]) in exact, not asymptotic, formulation.Translated from Problemy Ustoichivosti Stokhasticheskikh Modelei, pp. 41–47, 1981.     

Selfdecomposable Measures on Simply Connected Nilpotent Groups

The concepts of class L measures and selfdecomposable measures are generalised from vector spaces to simply connected nilpotent groups G. It has been shown that any full class L (probability) measure on G can be decomposed as = ₁* ... * _n, where each _i is a selfdecomposable measure on a subgroup G _i; itself is selfdecomposable under certain additional conditions—for example, when is symmetric. This generalizes a well known result on vector spaces. Some examples of class L measures on G are also constructed.