An extension of Hawkes theorem on the Hausdorff dimension of a Galton–Watson tree

Let ? be the genealogical tree of a supercritical multitype Galton–Watson process, and let Λ be the limit set of ?, i.e., the set of all infinite self-avoiding paths (called ends) through ? that begin at a vertex of the first generation. The limit set Λ is endowed with the metric d(ζ, ξ) = 2 ⁻ⁿ where n = n(ζ, ξ) is the index of the first generation where ζ and ξ differ. To each end ζ is associated the infinite sequence Φ(ζ) of types of the vertices of ζ. Let Ω be the space of all such sequences. For any ergodic, shift-invariant probability measure μ on Ω, define Ω_μ to be the set of all μ-generic sequences, i.e., the set of all sequences ω such that each finite sequence v occurs in ω with limiting frequency μ(Ω(v)), where Ω(v) is the set of all ω′?Ω that begin with the word v. Then the Hausdorff dimension of Λ∩Φ⁻¹ (Ω_μ) in the metric d is

The second centralizer of a Bernoulli shift is just its powers

If ℐ is a collection of measure preserving transformations of a probability space, byC(ℐ), the centralizer of ℐ, we mean the group of all measure preserving transformationsS such thatTS=ST for allT ∈ ℐ. We show here that ifT is a Bernoulli shift, thenC(C(T))={T ⁱ |i ∈ Z}. The proof is carried out by constructing an action of Z², {T ₁ ⁱ °T ₂ ⁱ |i, j ∈ Z}, whereT ₁ is a Bernoulli shift of arbitrary entropy, but for anyj ≠ 0,C({T ₁,T ₂ ^i} ={T ₁ ⁱ °T ₂ ^k l, k ∈ Z}. The construction is a two-dimensional analogue of Ornstein’s “rank one mixing” transformation.     

Cramér-type conditions and quadratic mean differentiability

Summary Let (Ω,A) be a measurable space, let Θ be an open set inR ^k , and let {P _θ; θ∈Θ} be a family of probability measures defined onA. Let μ be a σ-finite measure onA, and assume thatP _θ≪μ for each θ∈Θ. Let us denote a specified version ofdP _θ /d _μ byf(ω; θ). In many large sample problems in statistics, where a study of the log-likelihood is important, it has been convenient to impose conditions onf(ω; θ) similar to those used by Cramér [2] to establish the consistency and asymptotic normality of maximum likelihood estimates. These are of a purely analytical nature, involving two or three pointwise derivatives of lnf(ω; θ) with respect to θ. Assumptions of this nature do not have any clear probabilistic or statistical interpretation. In [10], LeCam introduced the concept of differentially asymptotically normal (DAN) families of distributions. One of the basic properties of such a family is the form of the asymptotic expansion, in the probability sense, of the log-likelihoods. Roussas [14] and LeCam [11] give conditions under which certain Markov Processes, and sequences of independent identically distributed random variables, respectively, form DAN families of distributions. In both of these papers one of the basic assumptions is the differentiability in quadratic mean of a certain random function. This seems to be a more appealing type of assumption because of its probabilistic nature. In this paper, we shall prove a theorem involving differentiability in quadratic mean of random functions. This is done in Section 2. Then, by confining attention to the special case when the random function is that considered by LeCam and Roussas, we will be able to show that the standard conditions of Cramér type are actually stronger than the conditions of LeCam and Roussas in that they imply the existence of the necessary quadratic mean derivative. The relevant discussion is found in Section 3. This research was supported by the National Science Foundation, Grant GP-20036.     

A quantum spin system with random interactions I

We study a quantum spin glass as a quantum spin system with random interactions and establish the existence of a family of evolution groups {τ_t(ω)}_ω∈/Ω of the spin system. The notion of ergodicity of a measure preserving group of automorphisms of the probability space Ω, is used to prove the almost sure independence of the Arveson spectrum Sp(τ(ω)) of τ_t(ε). As a consequence, for any family of (τ(ω),β) — KMS states {ρ(ω)}, the spectrum of the generator of the group of unitaries which implement τ(ω) in the GNS representation is also almost surely independent of ω.     

Estimates of the weak distance between finite-dimensional Banach spaces

We prove a variant of a theorem of N. Alon and V. D. Milman. Using it we construct for everyn-dimensional Banach spacesX andY a measure space Ω and two operator-valued functionsT: Ω→L(X, Y),S: Ω→L(Y, X) so that ∫_Ω S(ω)oT(ω)dω is the identity operator inX and ∫_Ω||S(ω)||·||T(ω)||dω=O(n ^α ) for some absolute constantα<1. We prove also that any subset of the unitn-cube which is convex, symmetric with respect to the origin and has a sufficiently large volume possesses a section of big dimension isomorphic to ak-cube. Research supported in part by a grant of the Israel Academy of Sciences.     

Estimates of the Bergman kernel for some pseudoconvex domains

Denote by K_ω(z, ζ) the Bergman kernel of a pseudoconvex domain Ω. For some classes of domains Ω, a relationship is found between the rate of increase of K_ω(z, z) as z tends to ∂Ω, and a purely geometric property of Ω. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 222, 1995, pp. 222–245.     

Generalized Bernstein Polynomials

We introduce polynomials B ⁿ _i (x;ω|q), depending on two parameters q and ω, which generalize classical Bernstein polynomials, discrete Bernstein polynomials defined by Sablonnière, as well as q-Bernstein polynomials introduced by Phillips. Basic properties of the new polynomials are given. Also, formulas relating B ⁿ _i (x;ω|q), big q-Jacobi and q-Hahn (or dual q-Hahn) polynomials are presented. This revised version was published online in July 2006 with corrections to the Cover Date.     

Polynomial Birth–Death Distribution Approximation in the Wasserstein Distance

The polynomial birth–death distribution (abbreviated, PBD) on ℐ={0,1,2,…} or ℐ={0,1,2,…,m} for some finite m introduced in Brown and Xia (Ann. Probab. 29:1373–1403, 2001) is the equilibrium distribution of the birth–death process with birth rates {α _i } and death rates {β _i }, where α _i ≥0 and β _i ≥0 are polynomial functions of i∈ℐ. The family includes Poisson, negative binomial, binomial, and hypergeometric distributions. In this paper, we give probabilistic proofs of various Stein’s factors for the PBD approximation with α _i =a and β _i =i+bi(i−1) in terms of the Wasserstein distance. The paper complements the work of Brown and Xia (Ann. Probab. 29:1373–1403, 2001) and generalizes the work of Barbour and Xia (Bernoulli 12:943–954, 2006) where Poisson approximation (b=0) in the Wasserstein distance is investigated. As an application, we establish an upper bound for the Wasserstein distance between the PBD and Poisson binomial distribution and show that the PBD approximation to the Poisson binomial distribution is much more precise than the approximation by the Poisson or shifted Poisson distributions.      

Equivalences between bloch type spaces and BOM spaces on the hyperbolic domains inC

For an analytic function f on the hyperbolic domain Ω inC, the following conclusions are obtained: (i) f∈B(Ω)=BMO A(Ω,m) if and only ifRef∈B_h(Ω)=BMOH(Ω,m). (ii) QB_h(Ω)=B_h(Ω)(BMOH _n(Ω,m)=BMOH(Ω,m)) if and only ifC(Ω)=inf{λ_Ω(z)·δ_Ω(z):z∈Ω}>0. Also, some applications to automorphic functions are considered. This research was supported by the Doctoral Program Foundation of Institute of Higher Education.     

Frobenius Extensions and Tilting Complexes

Let n ≥ 1 be an integer and π a permutation of I = {1, ⋯ ,n}. For any ring R, we provide a systematic construction of rings A which contain R as a subring and enjoy the following properties: (a) 1 = ∑  _i ∈ I e _i with the e _i orthogonal idempotents; (b) e _i x = xe _i for all i ∈ I and x ∈ R; (c) e _i A e _j  ≠ 0 for all i, j ∈ I; (d) e _i A _A  ≇ e _j A _A unless i = j; (e) every e _i Ae _i is a local ring whenever R is; (f) e _i A _A  ≅ Hom _R (Ae _π(i),R _R ) and _A Ae _π(i) ≅  _A Hom _R (e _i A, _R R) for all i ∈ I; and (g) there exists a ring automorphism η ∈ Aut(A) such that η(e _i ) = e _π(i) for all i ∈ I. Furthermore, for any nonempty π-stable subset J of I, the mapping cone of the multiplication map is a tilting complex. Dedicated to Takeshi Sumioka on the occasion of his 60th birthday.     

Green functions and eigenfunction expansions for the square of the Laplace–Beltrami operator on plane domains

For a certain class of domains Ω⊂ℂ with smooth boundary and Δtilde;_Ω=w ²Δ the Laplace–Beltrami operator with respect to the Poincaré metric ds ²=w(z)^-2 dz dz on Ω, we (1) show that the Green function for the biharmonic operator Δtilde;_Ω ², with Dirichlet boundary data, is positive on Ω×Ω; and (2) obtain an eigenfunction expansion for the operator Δtilde;_Ω, which reduces to the ordinary non-Euclidean Fourier transform of Helgason for Ω=𝔻 (the unit disc). In both cases the proofs go via uniformization, and in (1) we obtain a Myrberg-like formula for the corresponding Green function. Finally, the latter formula as well as the eigenfunction expansion are worked out more explicitly in the simplest case of Ω an annulus, and a result is established concerning the convergence of the series ∑ _ω∈G (1-|ω0|²) ^s for G the covering group of the uniformization map of Ω and 0<s<1. Received: August 21, 2000?Published online: October 30, 2002 RID="*" ID="*"The first author was supported by GA AV CR grants no. A1019701 and A1019005.     

Spectral structure of Anderson type Hamiltonians

We study self adjoint operators of the form?H _ω = H ₀ + ∑λ_ω(n) <δ _n ,·>δ _n ,?where the δ _n ’s are a family of orthonormal vectors and the λ_ω(n)’s are independently distributed random variables with absolutely continuous probability distributions. We prove a general structural theorem saying that for each pair (n,m), if the cyclic subspaces corresponding to the vectors δ _n and δ _m are not completely orthogonal, then the restrictions of H _ω to these subspaces are unitarily equivalent (with probability one). This has some consequences for the spectral theory of such operators. In particular, we show that “well behaved” absolutely continuous spectrum of Anderson type Hamiltonians must be pure, and use this to prove the purity of absolutely continuous spectrum in some concrete cases. Oblatum 27-V-1999 & 6-I-2000?Published online: 8 May 2000     

Some results on the majorization theorem of connected graphs

Let π = (d ₁, d ₂, ..., d _n ) and π′ = (d′ ₁, d′ ₂, ..., d′ _n ) be two non-increasing degree sequences. We say π is majorizated by π′, denoted by π ⊲ π′, if and only if π ≠ π′, Σ _i=1 ⁿ d _i = Σ _i=1 ⁿ d′ _i , and Σ _i=1 ^j d _i ≤ Σ _i=1 ^j d′ _i for all j = 1, 2, ..., n. Weuse C _π to denote the class of connected graphs with degree sequence π. Let ρ(G) be the spectral radius, i.e., the largest eigenvalue of the adjacent matrix of G. In this paper, we extend the main results of [Liu, M. H., Liu, B. L., You, Z. F.: The majorization theorem of connected graphs. Linear Algebra Appl., 431(1), 553–557 (2009)] and [Bıyıkoğlu, T., Leydold, J.: Graphs with given degree sequence and maximal spectral radius. Electron. J. Combin., 15(1), R119 (2008)]. Moreover, we prove that if π and π′ are two different non-increasing degree sequences of unicyclic graphs with π ⊲ π′, G and G′ are the unicyclic graphs with the greatest spectral radii in C _π and C′ _π , respectively, then ρ(G) < ρ(G′).     

The Helmholtz decomposition of weightedLq-spaces for Muckenhoupt weights

We consider weights of Muckenhoupt classA _q, 1<q<∞. For a bounded Lipschitz domain Ω⊂ℝⁿ we prove a compact embedding and a Poincaré inequality in weighted Sobolev spaces. These technical tools allow us to solve the weak Neumann problem for the Laplace equation in weighted spaces on ℝⁿ, ℝⁿ ₊, on bounded and on exterior domains Ω with boundary of classC ¹, which will yield the Helmholtz decomposition ofL _ω ^q(Ω)ⁿ for general ω∈A _q. This is done by transferring the method of Simader and Sohr [4] to the weighted case. Our result generalizes a result of Farwig and Sohr [2] where the Helmholtz decomposition ofL _ω ^p(Ω)ⁿ is proved for an exterior domain and weights of Muckenhoupt class without singularities or degeneracies in a neighbourhood of ϖΩ.

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Sunto In questo lavoro consideriamo dei pesi della classe di MuckenhouptA _q, 1<q<∞. Per un dominio limitato lipschitziano Ω⊂ℝⁿ, dimostriamo una immersione compatta ed una disuguaglianza di Poincaré in spazi di Sobolev con peso. Questa tecnica ci consente di risolvere il problema debole di Neumann per l’equazione di Laplace in spazi pesati in ℝⁿ, ℝⁿ ₊ in domini limitati ed in domini esterni con frontiera di classeC ¹, che conduce alla decomposizione di Helmholtz diL _ω ^q(Ω)ⁿ per un qualsiasi ω∈A _q. Il risultato è ottenuto trasferendo il metodo di Simader e Sohr [4] al caso pesato. Quello qui presente estende un risultato di Farwig e Sohr [2] dove la decomposizione di Helmholtz diL _ω ^q(Ω)ⁿ è dimostrata per domini esterni e pesi della classe di Muckenhoupt privi di singolarità in un intorno di ϖΩ.

     

Some divisibilities amongst the terms of linear recurrences

Let (u_n)n≥0 be a non-degenerate linear recurrence sequence of integers. We show that the set of positive integersn such that either ω)(n) orΩ(n) dividesu _n is of asymptotic density zero, where ω(n) and Ω(n) are the numbers of prime and prime power divisors ofn, respectively. The same also holds for the set of positive integersn such that τ(n)u _n , where τ(n) is the number of the positive integer divisors of n, provided thatu _n satisfies some mild technical conditions.     

On representation theorem of <Emphasis Type="Italic">G</Emphasis>-expectations and paths of <Emphasis Type="Italic">G</Emphasis>-Brownian motion

We give a very simple and elementary proof of the existence of a weakly compact family of probability measures {Pθ ： θ∈θ} representing an important sublinear expectation- G-expectation E[·]. We also give a concrete approximation of a bounded continuous function X（ω） by an increasing sequence of cylinder functions Lip（Ω） in order to prove that Cb（Ω） belongs to the completion of Lip（Ω） under the natural norm E[｜·｜].     

Hardy spaces via distribution spaces

Let ℐ(ℝⁿ) be the Schwartz class on ℝⁿ and ℐ_∞(ℝⁿ) be the collection of functions ϕ ∊ ℐ(ℝⁿ) with additional property that

An information geometry algorithm for distribution control

In this paper, we consider the problem of distribution control from the viewpoint of information geometry. Different from most existing models used in stochastic control, it is assumed that the control input directly affects the distribution of the system output in probability sense. Here, we set up a new manifold (S), meanwhile the B-spline manifold (B) and the system output manifold (M) can be referred to as its submanifolds. We give an information geometrical algorithm which can be called as geodesic-projection algorithm using the properties of manifold. In the geodesic step, we can obtain the geodesic equation from the initial point V₀ = (ω₁₀, ω₂₀, ··· , ω_(n−1)0) to the specified point V_g = (ω_1g, ω_2g, ··· , ω_(n−1)g) in B. This gives us an optimal trajectory for the points changing along in B. In the projection step, we project the sample points selected from the geodesic onto M. The coordinates of the projections in M give the trajectory of the control input u.     

Evasion and prediction

 Say that a function π:n ^<ω →n (henceforth called a predictor) k-constantly predicts a real xn ^ω if for almost all intervals I of length k, there is iI such that x(i)=π(x↾i). We study the k-constant prediction number v _n ^const (k), that is, the size of the least family of predictors needed to k-constantly predict all reals, for different values of n and k, and investigate their relationship. Received: 27 June 2001 / Revised version: 10 September 2001 / Published online: 10 October 2002 RID="*" ID="*" Supported by Grant–in–Aid for Scientific Research (C)(2)12640124, Japan Society for the Promotion of Science RID="†" ID="†" Supported by The Israel Science Foundation founded by the Israel Academy of Sciences and Humanities. Publication 762     

Modules of vector fields,differential forms and degenerations of differential systems

We deal with (n−1)-generated modules of smooth (analytic, holomorphic) vector fieldsV=(X ₁,..., X_n−1) (codimension 1 differential systems) defined locally on ℝ ⁿ or ℂ ⁿ , and extend the standard duality(X ₁,..., X_n−1)↦(ω), ω=Ω(X₁,...,X_n−1,.,) (Ω−a volume form) betweenV′s and 1-generated modules of differential 1-forms (Pfaffian equations)—when the generatorsX _i are linearly independent—onto substantially wider classes of codimension 1 differential systems. We prove that two codimension 1 differential systemsV and are equivalent if and only if so are the corresponding Pfaffian equations (ω) and provided that ω has1-division property: ωΛμ=0, μ—any 1-form ⇒ μ=fω for certain function germf. The 1-division property of ω turns out to be equivalent to the following properties ofV: (a)fX∈V, f—not a 0-divisor function germ ⇒X∈V (thedivision property); (b) (V ^⊥)^⊥=V; (c)V ^⊥=(ω); (d) (ω)^⊥=V, where ⊥ denotes the passing from a module (of vector fields or differential 1-forms) to its annihilator. Supported by Polish KBN grant N^o 2 1090 91 01. Partially supported by the fund for the promotion of research at the Technion, 100–942.