On the asymptotic and invariantσ-algebras of random walks on locally compact groups

Summary Consider a random walk of law on a locally compact second countable groupG. Let the starting measure be equivalent to the Haar measure and denote byQ the corresponding Markov measure on the space of pathsG . We study the relation between the spacesL (G , ^a ,Q) andL (G , ⁱ ,Q) where ^a and ⁱ stand for the asymptotic and invariant -algebras, respectively. We obtain a factorizationL (G , ^a ,Q) L (G , ⁱ ,Q)L (C) whereC is a cyclic group whose order (finite or infinite) coincides with the period of the Markov shift and is determined by the asymptotic behaviour of the convolution powers ⁿ.     

On the complete chromatic number of Halin graphs

ThisresearchissupportedbytheNationalNaturalScienceFoundationofChina.Write.1.IntroductionDefinition1.FOrany3-connectedplanargraphG(V,E,F)withA(G)23,iftheboundaryedgesoffacefowhichisadjacenttotheothersareremoved,itbecomesatree,andthedegreeofeachvertexofV(fo)is3,andthenGiscalledaHalingraph;foiscalledtheouterfaceofG,andtheotherscalledtheinteriorfaces,thevenicesonthefacefoarecalledtheoutervenices,theoillersarecalledtheinterior...ti..,tll.ForanyplanargraphG(V,E,F),f,f'eF,fisadjacenttof'ifan…     

Spitzer's condition and ladder variables in random walks

Summary Spitzer's condition holds for a random walk if the probabilities _n =P{ _n > 0} converge in Cèsaro mean to , where 0<<1. We answer a question which was posed both by Spitzer [12] and by Emery [5] by showing that whenever this happens, it is actually true that n converges to . This also enables us to give an improved version of a result in Doney and Greenwood [4], and show that the random walk is in a domain of attraction, without centering, if and only if the first ladder epoch and height are in a bivariate domain of attraction.     

Subsets with Restricted Movement

Let G be a finite permutation group on a set with no fixed points in and let m and k be integers with 0 < m < k. For a finite subset of the movement of is defined as move() = max_gG| ^g \ |. Suppose further that G is not a 2-group and that p is the least odd prime dividing |G| and move() m for all k-element subsets of . Then either || k + m or k (7m – 5) / 2, || (9m – 3)/2. Moreover when || > k + m, then move() m for every subset of .     

Polynomially convex orbits of compact lie groups

LetV be a finite dimensional complex linear space and letG be a compact subgroup of GL(V). We prove that an orbitG, V, is polynomially convex if and only ifG is closed andG is the real form ofG . For every orbitG which is not polynomially convex we construct an analytic annulus or strip inG with the boundary inG. It is also proved that the group of holomorphic automorphisms ofG which commute withG acts transitively on the set of polynomially convexG-orbits. Further, an analog of the Kempf-Ness criterion is obtained and homogeneous spaces of compact Lie groups which admit only polynomially convex equivariant embeddings are characterized.Supported by Federal program Integratsiya, no. 586.Supported by INTAS grant 97/10170.     

Partially ordered permutation groups

In this note we discuss permutation groups (G, ) in which the set admits aG-invariant order. By aG-invariant partial order (G-partial order) we mean a partial order < of such that < implies g<g, for all and in andg inG. If the set admits aG-partial order which is a total order, then (G, ) is an O-permutation group (orderable permutation group).The main concern of this paper is the development of a foundation for partially ordered permutation groups analogous to the existing one for partially ordered groups, as found in Fuchs [2].     

Stochastic integrals on general topological measurable spaces

Summary A general theory of stochastic integral in the abstract topological measurable space is established. The martingale measure is defined as a random set function having some martingale property. All square integrable martingale measures constitute a Hilbert space M ². For each M ², a real valued measure on the predictable -algebra is constructed. The stochastic integral of a random function with respect to is defined and investigated by means of Riesz's theorem and the theory of projections. The stochastic integral operator I is an isometry from L ²() to a stable subspace of M ², its inverse is defined as a random Radon-Nikodym derivative. Some basic formulas in stochastic calculus are obtained. The results are extended to the cases of local martingale and semimartingale measures as well.     

Asymptotics of the generating function for the volume of the Wiener sausage

Summary We consider the generating function of the voltime of the Wiener sausageC (t), which is the -neighbourhood of the Wiener path in the time interval [0,t]. For <0, the limiting behavior fort, up to logarithmic equivalence, had been determined in a celebrated work of Donsker and Varadhan. For >0 it had been investigated by van den Berg and Tóth, but in contrast to the case <0, there is no simple expression for the exponential rate known. We determine the asymptotic behaviour of this rate for small and large .     

On certain quotients of Hardy spaces

The quotient space of a Hardy space on a half-plane Imz> modulo the subspace of elements containing a factore ^iz is in some sense independent of . A formula is derived which exhibits a correspondence between any two such quotient spaces.Supported by the Göran Gustafsson Foundation for Research in Natural Sciences and Medicine.     

Blow-up rates for parabolic systems

Let ⁿ be a bounded domain andB _R be a ball in ⁿ of radiusR. We consider two parabolic systems: ut=u +f(), i= +g(u) in × (0,T) withu=v=0 on × (0,T) andu _t =u, v _t =v inB _r × (0,T) withe/v=f (v), e/v=g(u) onB _R × (0,T). Whenf(v) andg(u) are power law or exponential functions, we establish estimates on the blow-up rates for nonnegative solutions of the systems.     

Covering a Finite Abelian Group by Subset Sums

Let G be an abelian group of order n. The critical number c(G) of G is the smallest s such that the subset sums set (S) covers all G for eachs ubset SG\{0} of cardinality |S|s. It has been recently proved that, if p is the smallest prime dividing n and n/p is composite, then c(G)=|G|/p+p–2, thus establishing a conjecture of Diderrich.We characterize the critical sets with |S|=|G|/p+p–3 and (S)=G, where p3 is the smallest prime dividing n, n/p is composite and n7p²+3p.We also extend a result of Diderrichan d Mann by proving that, for n67, |S|n/3+2 and S=G imply (S)=G. Sets of cardinality for which (S) =G are also characterized when n183, the smallest prime p dividing n is odd and n/p is composite. Finally we obtain a necessary and sufficient condition for the equality (G)=G to hold when |S|n/(p+2)+p, where p5, n/p is composite and n15p².* Work partially supported by the Spanish Research Council under grant TIC2000-1017 Work partially supported by the Catalan Research Council under grant 2000SGR00079     

Quenched invariance principles for walks on clusters of percolation or among random conductances

In this work we principally study random walk on the supercritical infinite cluster for bond percolation on ^d. We prove a quenched functional central limit theorem for the walk when d4. We also prove a similar result for random walk among i.i.d. random conductances along nearest neighbor edges of ^d, when d1.V. Sidoravicius would like to thank the FIM for financial support and hospitality during his multiple visits to ETH. His research was also partially supported by FAPERJ and CNPq.     

On pathwise behavior of multiserver queues

In this paper, by exploiting recent results on the pathwise behavior of the workload process in single server, work conserving queues of theG/G/1/ type, we show that the workload of multiserver, work conserving queues ofG/G/m/ (m<) (andG/G/) queues satisfies an o(t) growth condition, provided that the time average of the work brought into the system is less thanm form < (and finite form=).     

On Singularities of Generating Functions of Pólya Frequency Sequences of Finite Order

Let E be a closed set in C satisfying the conditions: (i) E is symmetric with respect to the real axis, (ii) E {z:|z|1} = . For any r 1 there exists a function f(z) satisfying the properties: (i) f(z) is a generating function of a Pólya frequency sequence of order r, (ii) the singularity set of f(z) is E {1}.     

The martin boundary for harmonic functions on groups of automorphisms of a homogeneous tree

Consider a closed subgroup of the automorphism group of a homogeneous treeT, and assume that acts transitively on the vertex set. Suppose that is a probability measure on which has continuous density with respect to Haar measure and whose support is compact open and generates as a closed semigroup. It is shown that the Martin boundary of with respect to the random walk with law coincides with the space of ends ofT. This extends known results for free groups and applies, for example, to the affine group over a non archimedean local field.     

Future independent times and Markov chains

Summary A random timeT is a future independent time for a Markov chain (X _n ) ₀ ifT is independent of (X _T+n ) _n ^/ =0 and if (X _T+n ) _n ^/ =0 is a Markov chain with initial distribution and the same transition probabilities as (X _n ) ₀ . This concept is used (with the conditional stationary measure) to give a new and short proof of the basic limit theorem of Markov chains, improving somewhat the result in the null-recurrent case.This work was supported by the Swedish Natural Science Research Council and done while the author was visiting the Department of Statistics, Stanford University     

Lattices on Nonuniform Trees

Let X be a locally finite tree, and let G=Aut(X). Then G is a locally compact group. We show that if X has more than one end, and if G contains a discrete subgroup such that the quotient graph of groups \\X is infinite but has finite covolume, then G contains a nonuniform lattice, that is, a discrete subgroup such that \G is not compact, yet has a finite G-invariant measure.     

Stochastic models of homogeneity

The probabilityp()=P{–<) is used as a characteristic of homogeneity of the system (, ) of independent identically distributed random variables. We consider some estimation methods of this characteristic and a method of classification of observations minimizing the between-class dissimilarity measure.Translated from Statisticheskie Metody Otsenivaniya i Proverki Gipotez, pp. 120–123, 1986.     

Extremes of a Gaussian Process and the Constant H α

Consider a triangular array of standard Gaussian random variables {_n,i, i 0, n 1} such that {_n,i, i 0} is a stationary normal sequence for each n 1. Let _n,k = corr(_n,i,_n,i+k). If (1-_n,k)log n _k (0,) as n for some k, then the locations where the extreme values occur cluster and the limiting distribution of the maxima is still the Gumbel distribution as in the stationary or i.i.d. case, but shifted by a parameter measuring the clustering. Triangular arrays of Gaussian sequences are used to approximate a continuous Gaussian process X(t), t 0. The cluster behavior of the random sequence refers to the behavior of the extremes values of the continuous process. The relation is analyzed. It reveals a new definition of the constants H used for the limiting distribution of maxima of continuous Gaussian processes and provides further understanding of the limit result for these extremes.     

On the Heyde Theorem for Finite Abelian Groups

It is well-known Heyde's characterization theorem for the Gaussian distribution on the real line: if _j are independent random variables, _j , _j are nonzero constants such that _i ± _j ^–1 _j 0 for all i j and the conditional distribution of L ₂=_{1 1} + ··· + _{n n} given L ₁=_{1 1} + ··· + _{n n} is symmetric, then all random variables _j are Gaussian. We prove some analogs of this theorem, assuming that independent random variables take on values in a finite Abelian group X and the coefficients _j , _j are automorphisms of X.