Selfdecomposable distributions for maxima of independent random vectors

Summary In the present paper the limit laws for conveniently normalized multivariate sample extremes are characterized by means of the decomposability of probability distributions. Continuous automorphisms ofR ^d =[–,]^d with respect to the operation v defined by x y=(max(x _i, y_i),i=1... d) are treated as norming mappings. An integral representation of the limit distributions is found using their log-concavity and a decomposition ofR ^d in orbits of the norming family. Finally an example is given as an illustration.Research supported in part by the Committee of Science, Bulgarian Concil of Ministers, under contract no. 60/1987     

Unique determination of the convolutions of measures in Rm,m⩾2, by their restriction to a set

Restrictions are indicated on a complex-valued measure in the spaceR ^m ,m 2., under which the n-fold convolution ^n*,n 2, is uniquely determined by its values on any semispacex ₁<r, rR.Translated from Teoriya Funktsii, Funktsional'nyi Analiz i Ikh Prilozheniya, No. 50, pp. 86–90, 1988.     

Submodule lattice quasivarieties and exact embedding functors for rings with prime power characteristic

Given ringsR with prime power characteristicp ^k , quasivarieties (R) of lattices generated by lattices of submodules ofR-modules are studied. An algebra of expressionsd not dependent onR is developed, such that each suchd uniquely determines a two-sides ideald _R ofR. The main technical result is that (R) (S) makes all implications of the formd _s =S d_R=R true, for any such expressiond. The proof makes use of the known equivalence between (R) (S) and existence of an exact embedding functorR-Mod S -Mod. Fork 2, the ordered setW(p ^k ) of all lattice quasivarieties (R),R having characteristic p _K , is shown to be large and complicated, with ascending and descending chains and antichains having continuously many elements. More precisely,W(p ^k ) has a subset which is order isomorphic to the Boolean algebra of all subsets of a denumerably infinite set. Also, given any prime powerp ^k ,k 2, a ringR can be constructed so that (R) and (R ^op) for the opposite ringR ^op are distinct elements ofW(p ^k ).Presented by R. Freese.Research partially supported by Hungarian National Foundation for Scientific Research grant no. 1903.     

Fundamental equation of information revisited

Summary This paper presents a new, shorter and more direct proof of the following result of J. Aczél and C. T. Ng: IfM: J R (J =]0, 1[ ^k ) is both multiplicative and additive, then the general solution: J R of(x) + M(1 – x)(y/1 – x) = (y) + M(1 – y)(x/1 – y) (x, y, x + y J) is given by(x) = ifM = 0,(x) = M(x)[L(x) + ] + M(1 – x)L(1 – x) ifM 0,where is an arbitrary constant andL: J R is an arbitrary solution of the logarithmic functional equationL(xy) = L(x) + L(y) (x, y J). Also, some extensions of this result to fields more general than the reals are given.     

Estimates of Green Function for Relativistic α-Stable Process

We call an R ^d -valued stochastic process X _t with characteristic function exp{–t{(m ^2/+²)^/2–m}},R ^d ,m>0, the relativistic -stable process. In the paper we derive sharp estimates for the Green function of the relativistic -stable process on C ^1,1 domains. Using these estimates we provide lower and upper bounds for the Poisson kernel. As another application we derive 3G Theorem and Boundary Harnack Principle for C ^1,1 domains.     

Representation of linearly additive random fields

Summary Chentsov type representation theorem is proved for stochastically continuous, linearly additive, infinitely divisible random field without Gaussian component, where a random fieldX={X(t), tR ^d } is called linearly additive if the stochastic process defined by ()=X(a+b), R, has independent increments for every pair(a, b), a, bR ^d . In passing it is shown that there exists a natural one-to-one correspondence between stochastically continuous, linearly additive Poisson random fields onR ^d and locally finite, bundleless measures on the space of all (d-1)-hyperplanes inR ^d . The latter result is closely related to Ambartzumian's theorem on the representation of linearly additive pseudometrics in the plane.     

Can you cover your shadows?

A subsetX of thed-dimensional Euclidean space ^d can cover its shadows inR ^d , if every orthogonal projection ofX onto a (d–1)-dimensional linear subspace of ^d is contained in some congruent copy ofX. Whereas every two-dimensional convex discC R ^d has this property, no (d–1)-polytope does, provided thatd>-4.     

Asymptotic expansion of the spectral function of elliptic operators inR n

Let {E} be the spectral family of the operator corresponding to a self-adjoint second-adjoint second-order elliptic differential equation or to a first-order elliptic system inR ⁿ . The coefficients of the equations are assumed to be constant in a neighborhood of infinity and trapping rays are absent. We obtain the full asymptotic expansion as of the kernel e ¹ (x, y) of the operator dE/d in terms of rays going from the point y into the point x.Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 12, pp. 75–87, 1987.     

Variation of Bergman metrics on Riemann surfaces

We study the variation of the Berman metric K(t,)|d|² on the Riemann surface R(t) (which varies with complex parameter t in a disk B) such that :B is a 2-dimensional Stein space with ^–1(t)=R(t),t B. We show that log K(t,)|d|² is plurisubharmonic in , and apply it to give the uniformization condition for the special Stein space.Revised version: 18 February 2004Mathematics Subject Classification (2000): 30C85, 30F15, 32F05, 32H10     

Exponential estimates for the Wiener sausage

Summary LetC (t) be the Wiener sausage of radius inR ^d up to timet. We obtain bounds on the asymptotics ofE exp (|C (t)|) ast, for all >0.     

Almost-Periodic Multipliers

In this work we study the necessary and sufficient conditions for a generalized trigonometric series in order for it to be the series of a Stepanoff almost-periodic function fS ^q (R),1q<. We consider analogous conditions for functions belonging to D(,R). Finally, we characterize the multipliers of invariance of the (B ¹(),B ¹()) type.     

Isotropic random walks in a tree

Summary Let T be an infinite homogeneous tree of order a+1. We study Markov chains {X _n} in T whose transition functions p(x, y)=A[d(x,y)] depend only on the shortest distance between x and y in the graph. The graph T can be represented as a symmetric space of a p-adic matrix group; we prove a series of results using essentially the spherical functions of this symmetric space. Theorem 1. d(X _n,x) n a.s., where >0 if A(0) 1, X ₀=x. Assuming {X _n} is strongly aperiodic, Theorem 2. p ₂(x, y)CRⁿ/n^3/2 for fixed x, y where R=(d) A(d)<1, and if E[d(X₁, X₀)²]<, Theorem 3. R(1–u, x, y) = (1–u)ⁿp_n(x, y)=Ca^–d[exp(–du/)+o_d(1)] as d=d(x,y) uniformly for 0u2. Using Theorem 3, we calculate the Martin boundary Dirichlet kernel of p(x, y) on T, which turns out to be independent of {itA(d)}. We also consider a stepping-stone model of a randomly-mating-and-migrating population on the nodes of T. If initially all individuals are distinct, then in generation n approximately half of the individuals of a given type are within n of a typical one and essentially all are within 2n.This work was partially supported by the National Science Foundation under grant number MCS 75-08098-A01For the academic year 1977–78: Department of Mathematics GN-50, University of Washington, Seattle, Washington 98195 USA     

Asymptotics of Dominated Gaussian Maxima

Let {X _n , n1} be a sequence of independent Gaussian random vectors in R ^d d2. In this paper an asymptotic evaluation of P{max₁in X _i a _n Z+b _n } with Z another Gaussian random vector is obtained for a _n, b _n R ^d two vectors obeying certain conditions.     

Time-Localization of Random Distributions on Wiener Space II: Convergence,Fractional Brownian Density Processes

For a random element X of a nuclear space of distributions on Wiener space C([0,1],R ^d ), the localization problem consists in projecting X at each time t[0,1] in order to define an S(R ^d )-valued process X={X(t),t[0,1]}, called the time-localization of X. The convergence problem consists in deriving weak convergence of time-localization processes (in C([0,1],S(R ^d )) in this paper) from weak convergence of the corresponding random distributions on C([0,1],R ^d ). Partial steps towards the solution of this problem were carried out in previous papers, the tightness having remained unsolved. In this paper we complete the solution of the convergence problem via an extension of the time-localization procedure. As an example, a fluctuation limit of a system of fractional Brownian motions yields a new class of S(R ^d )-valued Gaussian processes, the fractional Brownian density processes.     

On Embeddings of the Flag Geometries of Projective Planes in Finite Projective Spaces

The flag geometry =( ) of a finite projective plane of order s is the generalized hexagon of order (s, 1) obtained from by putting equal to the set of all flags of , by putting equal to the set of all points and lines of and where I is the natural incidence relation (inverse containment), i.e., is the dual of the double of in the sense of Van Maldeghem Mal:98. Then we say that is fully and weakly embedded in the finite projective space PG(d, q) if is a subgeometry of the natural point-line geometry associated with PG(d, q), if s = q, if the set of points of generates PG(d, q), and if the set of points of not opposite any given point of does not generate PG(d, q). Preparing the classification of all such embeddings, we construct in this paper the classical examples, prove some generalities and show that the dimension d of the projective space belongs to {6,7,8}.     

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.     

Anti-Wick Symbols for Infinite Products in K-Homology

We consider infinite products in K-homology. We study these products in relation with operators on filtered Hilbert spaces, and infinite iterations of universal constructions on C*-algebras. In particular, infinite tensor power of extensions of pseudodifferential operators on R are considered. We extend anti-Wick pseudodifferential operators to infinite tensor products of spaces of the type L ²(R), and compare our infinite tensor power construction with an extension of pseudodifferential operators on R . We show that the K-theory connecting maps coincide. We propose a natural definition of ellipticity for anti-Wick operators on R, compute the corresponding index, and draw some consequences concerning these operators.     

On infinitely divisible self-similar random fields

Summary A class of self-similar stationary random fields in ^d , d1 with finite variance is constructed by means of multiple stochastic integrals with respect to the Poisson random measure in ^d+1. Various topics associated with these fields such as subordination, ergodicity, existence of higher order moments, uniqueness of stochastic integral representation, renormalized powers of linear generalized fields and some limit theorems are studied. A Lévy-Hinin type formula for the characteristic functional of general infinitely divisible self-similar random fields with finite variance is obtained.     

A criterion for <Emphasis Type="Italic">C</Emphasis><Superscript>2</Superscript>-rectifiability of sets

The following result is proved: Let be a n-dimensional C¹-submanifold of R^N which is domain of a given _nR^N-valued map of class C¹. Then the set of all points P such that (P) is non-zero, simple and enveloped by T_P is C²-rectifiable. As a corollary we get a criterion for the C²-rectifiability of a rectifiable set based on the rectifiability of some generalized Gauss lift to the Grassmanian bundle R^N×G(N,n). Mathematics Subject Classification (2000) Primary 49Q15, 53A07; Secondary 49Q20, 49N60     

Über Mosers regularisiertes Variationsproblem für minimale Blätterungen desn-dimensionalen Torus

We consider regular and Cantor-like minimal foliations of the (n+1)-dimensional TorusT ⁿ⁺¹ whose leaves minimize a given variational integral. Each leaf of such a generalized foliation lies in the universal coveringR ⁿ⁺¹ within a finite distance to the affine leaves (z, x+) of fixed R ⁿ . We show that the conjugation-functionU (x,), mapping the affine leaves (x, x+) into the leaves(x,U (x,x+)) of the generalized foliation, is itself a minimal solution of an extended degenerate variational problem onT ⁿ +1. If R ⁿ /Q ⁿ the functionU is characterized in a unique way as (discontinuous) limit of the minimal solutions of the corresponding regularized problem.