Selective Limit Theorems for Random Walks on Parabolic Biangle and Triangle Hypergroups

Let K be respectively the parabolic biangle and the triangle in and be a sequence in [0, +[ such that lim_p (p)=+. According to Koornwinder and Schwartz,⁽⁷⁾ for each there exist a convolution structure (*_(p)) such that (K, *_(p)) is a commutative hypergroup. Consider now a random walk on (K, *_(p)), assume that this random walk is stopped after j(p) steps. Then under certain conditions given below we prove that the random variables on K admit a selective limit theorems. The proofs depend on limit relations between the characters of these hypergroups and Laguerre polynomials that we give in this work.     

The type set for some measures on \mathbb{R}^{2n} with n-dimensional support

Let be realhomogeneous functions in ofdegree and let bethe Borel measure on given by

On the speed of convergence in the central limit theorem of log-likelihood ratio processes

Let , the parameter space, be an open subset ofR ^k ,k1. For each , let the r.v.'sX _n ,n=1, 2,... be defined on the probability space (X, P ) and take values in (S,S,L) whereS is a Borel subset of a Euclidean space andL is the -field of Borel subsets ofS. ForhR _k and a sequence of p.d. normalizing matrices _n = _n ^{k × k} (₀ set _n ^* = ^* = ₀ + _n h, where ₀ is the true value of , such that ^*, . Let _n (^*, _*)( be the log-likelihood ratio of the probability measure with respect to the probability measure , whereP _n is the restriction ofP over _n = (X ₁,X ₂,...,X _n . In this paper we, under a very general dependence setup obtain a rate of convergence of the normalized log-likelihood ratio statistic to Standard Normal Variable. Two examples are taken into account.     

A Delocalization Property for Assembly Maps and an Application in Algebraic K-Theory of Group Rings

Let be a group, F the free -module on the set of finite order elements in , with acting by conjugation, and the ring extension of by % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeaacaGaaiaabeqaamaabaabaaGcbaWaaiWaaeaada% WcaaqaaiaaigdaaeaatCvAUfKttLearyGqLXgBG0evaGqbciab-5ga% UbaaieaacaGFLbGaaGOmaiaabc8acqWFPbqAcaqGVaGae8NBa42aaq% qaaeaacqGHdicjcqaHZoWzcqGHiiIZcqqHtoWrcaqGGaGaae4Baiaa% bAgacaqGGaGaae4BaiaabkhacaqGKbGaaeyzaiaabkhacaqGGaGae8% NBa4gacaGLhWoaaiaawUhacaGL9baaaaa!563E!\[\left\{ {\frac{1}{n}e2{\text{\pi }}i{\text{/}}n\left| {\exists \gamma \in \Gamma {\text{ of order }}n} \right.} \right\}\]. For a ring R with , we build an injective assembly map , detected by the Dennis trace map. This is proved by establishing a delocalization property for the assembly map in Hochschild homology, namely providing a gluing of simpler assembly maps (i.e. localized at the identity of ) to build , and by delocalizing a known assembly map in K-theory to define . We also prove the delocalization property in cyclic homology and in related theories.     

Beckman-Quarles type theorems for mappings from $$ \mathbb{R}^n $$ to $$ \mathbb{C}^n $$

Summary. Let We say that preserves the distance d 0 if for each implies Let A _n denote the set of all positive numbers d such that any map that preserves unit distance preserves also distance d. Let D _n denote the set of all positive numbers d with the property: if and then there exists a finite set S _xy with such that any map that preserves unit distance preserves also the distance between x and y. Obviously, We prove: (1) (2) for n 2 D _n is a dense subset of (2) implies that each mapping f from to (n 2) preserving unit distance preserves all distances, if f is continuous with respect to the product topologies on and      

Best Khintchine Type Inequalities for Sums of Independent,Rotationally Invariant Random Vectors

Let be an i.i.d. sequence of rotationally invariant random vectors in . If X ₁² is dominated (in the sense defined below) by Z² for a rotationally invariant normal random vector Z in , then for each k and

Differentiable structures with zero entropy on simply connected 4-manifolds

We show that a closed 4-dimensional simply connected topological manifoldM admits a differentiable structure with aC Riemannian metric whose geodesic flow has zero topological entropy if and only ifM is homeomorphic toS ⁴, ²,S ²×S ², or ²#².     

Limit theorems for compact two-point homogeneous spaces of large dimensions

Let be the field , , or of real dimension . For each dimensiond2, we study isotropic random walks(Y ₁)₁0 on the projective space with natural metricD where the random walk starts at some with jumps at each step of a size depending ond. Then the random variablesX ₁ ^d :=cosD(Y ₁ ^d ,x ₀ ^d ) form a Markov chain on [–1, 1] whose transition probabilities are related to Jacobi convolutions on [–1, 1]. We prove that, ford, the random variables (vd/2)(X _l(d) ^d +1) tend in distribution to a noncentral ²-distribution where the noncentrality parameter depends on relations between the numbers of steps and the jump sizes. We also derive another limit theorem for as well as thed-spheresS ^d ford.     

Interior points of the convex hull of few points in\mathbb{E}^d

We show that ifP , |P|=d+k,dk1 andO int convP, then there exists a simplexS of dimension with vertices inP, satisfyingO rel intS, the bound being sharp. We give an upper bound for the minimal number of vertices of facets of a (j-1)-neighbourly convex polytope in withv vertices.Research (partially) supported by Hung. Nat. Found. for Sci. Research, grant no. 1817Research (partially) supported by Hung. Nat. Found. for Sci. Research, grant no. 326-0213     

Krasnosel'skii numbers for bounded finitely starlike sets inR d

For eachk andd, 1kd, definef(d, d)=d+1 andf(d, k)=2d if 1kd–1. The following results are established:Let be a uniformly bounded collection of compact, convex sets inR ^d . For a fixedk, 1kd, dim {MM in }k if and only if for some > 0, everyf(d, k) members of contain a commonk-dimensional set of measure (volume) at least.LetS be a bounded subset ofR ^d . Assume that for some fixedk, 1kd, there exists a countable family of (k–l)-flats {H _i :i1} inR ^d such that clS S {H_i i 1 } and for eachi1, (clS S) H _i has (k–1) dimensional measure zero. Every finite subset ofS sees viaS a set of positivek-dimensional measure if and only if for some>0, everyf(d,k) points ofS see viaS a set ofk-dimensional measure at least .The numbers off(d,d) andf(d, 1) above are best possible.Supported in part by NSF grant DMS-8705336.     

The Minimal Subgroup of a Random Walk

It is proved that for each random walk (S _n ) _n0 on ^d there exists a smallest measurable subgroup of ^d , called minimal subgroup of (S _n ) _n0, such that P(S _n )=1 for all n1. can be defined as the set of all x ^d for which the difference of the time averages n ^{–1 n} _k=1 P(S _k ) and n ^{–1 n} _k=1 P(S _k +x) converges to 0 in total variation norm as n. The related subgroup * consisting of all x ^d for which lim _n P(S _n )–P(S _n +x)=0 is also considered and shown to be the minimal subgroup of the symmetrization of (S _n ) _n0. In the final section we consider quasi-invariance and admissible shifts of probability measures on ^d . The main result shows that, up to regular linear transformations, the only subgroups of ^d admitting a quasi-invariant measure are those of the form ₁×...× _k × ^l–k ×{0} ^d–l , 0kld, with ₁,..., _k being countable subgroups of . The proof is based on a result recently proved by Kharazishvili⁽³⁾ which states no uncountable proper subgroup of admits a quasi-invariant measure.     

Convergence of series of translations

For a mean zero norm one sequence (f _n )L ₂[0, 1], the sequence (f _n {nx+y}) is an orthonormal sequence inL ₂([0, 1]²); so if , then converges for a.e. (x, y)[0, 1]² and has a maximal function inL ₂([0, 1]²). But for a mean zerofL ₂[0, 1], it is harder to give necessary and sufficient conditions for theL ₂-norm convergence or a.e. convergence of . Ifc _n 0 and , then this series will not converge inL ₂-norm on a denseG subset of the mean zero functions inL ₂[0, 1]. Also, there are mean zerofL[0, 1] such that never converges and there is a mean zero continuous functionf with a.e. However, iff is mean zero and of bounded variation or in some Lip() with 1/2<1, and if |c _n | = 0(n ^–) for >1/2, then converges a.e. and unconditionally inL ₂[0, 1]. In addition, for any mean zerof of bounded variation, the series has its maximal function in allL _p[0, 1] with 1p<. Finally, if (f _n )L [0, 1] is a uniformly bounded mean zero sequence, then is a necessary and sufficient condition for to converge for a.e.y and a.e. (x _n )[0, 1]. Moreover, iffL [0, 1] is mean zero and , then for a.e. (x _n )[0, 1], converges for a.e.y and in allL _p [0, 1] with 1p<. Some of these theorems can be generalized simply to other compact groups besides [0, 1] under addition modulo one.     

Euler Characteristics of Local Systems on \mathcal{M}> 2

We calculate the Euler characteristics of the local systems S ^k S ² on the moduli space ₂ of curves of genus 2, where is the rank 4 local system R ¹ _* .     

Linear Networks and Convex Polytopes

It is shown that the set [G,] of immersed linear networks in that are parallel to a given immersed linear network and have the same boundary as is a convex polyhedral subset of the configuration space of movable vertices of the graph G. The dimension of [G,] is calculated, and the number of its maximal faces is estimated. As an application, the spaces of all locally minimal and weighted minimal networks with fixed boundary and topology in are described. Bibliography: 21 titles.     

More on limit theorems for iterates of probability measures on semigroups and groups

Summary In this paper, we continue earlier works of one of the authors on vague convergence of the sequence _k,n= _k+1 *...* _n, where _n is a sequence of probability measures on semigroups or groups. Typical results in this paper are: Theorem. Let S be a locally compact noncompact second countable group such that being the support of a probability measure on S. Suppose there exists an open set V with compact closure such that x ^–1 Vx=V for every xS. Then for all compact sets K, sup{ ⁿ (Kx): xS0 as n. Theorem. Let S be an at most countable discrete group. Let _n be a sequence of probability measures on S. Then for all nonnegative integers k, the sequence _k,n converges vaguely to some probability measure if and only if there exists a finite subgroup G such that the series and for any proper subgroup G of G and any choice of elements g_n in S, the series . A sufficient condition for the vague convergence of the sequence _k,n to a probability measure is that (i) there exists a finite subgroup G such that and (ii) _n(e)>s>0 for all n, e being the identity.The author was supported by NSF grant MCS77-03639     

An approach to the constructivization of Cantor's set theory

A new approach is proposed for the construction of constructive analogs of set theory in hyperarithmetic languages , where is a scale of constructive ordinals. For every ordinal in the language , a special relation of equality = is defined for codes of one-parameter formulas (conditions) of the level in a constructive hyperarithmetic hierarchy corresponding to the scale . The membership relation, (also expressible in the language ), is defined by the conditionx y=z(z= x&z y), where the relation is obtained by suitable refinement of the traditional representations of the constructive relation of membership. This results in a hierarchy of constructive analogsM of the theory of sets (in which the sets are represented by codes of conditions of level , identified modulo the relation =, and is taken as the relation of membership). Some properties of this hierarchy are introduced which show that for the limits ,M is sufficiently rich from the traditional set theoretic standpoint.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 68, pp. 38–49, 1977.     

Finitely starlike sets whose F-stars have positive measure

Let and assume that there is a countable collection of lines {L _i : 1 i} such that (int cl S) and ((int cl S) S) L _i has one-dimensional Lebesgue measure zero, 1 i. Then every 4 point subset ofS sees viaS a set of positive two-dimensional Lebesgue measure if and only if every finite subset ofS sees viaS such a set. Furthermore, a parallel result holds with two-dimensional replaced by one-dimensional. Finally, setS is finitely starlike if and only if every 5 points ofS see viaS a common point. In each case, the number 4 or 5 is best possible.Supported in part by NSF grant DMS-8705336.     

Singular perturbations in foliations

Summary We define a constraint system , [0,₀), which is a kind of family of vector fields on a manifold. This is a generalized version of the family of the equations , [0,₀),x ^m ,y ⁿ . Finally, we prove a singular perturbation theorem for the system , [0,₀).Dedicated to Professor Kenichi Shiraiwa on his 60th birthday     

On Distance-Regular Graphs with Height Two

Let be a distance-regular graph with diameter at least three and height h = 2, where . Suppose that for every in and in _d(), the induced subgraph on _d() ₂() is a clique. Then is isomorphic to the Johnson graph J(8, 3).     

On the Sobolev distance of convex bodies

Summary LetK ^d denote the cone of all convex bodies in the Euclidean spaceK ^d . The mappingK h _K of each bodyK K ^d onto its support function induces a metric _w onK ^d by" _w (K, L)h _L –h _{K w} where _w is the Sobolev I-norm on the unit sphere . We call _w (K, L) the Sobolev distance ofK andL. The goal of our paper is to develop some fundamental properties of the Sobolev distance.