Properly discontinuous actions of subgroups in amenable algebraic groups and its application to affine motions

This note will concern properly discontinuous actions of subgroups in real algebraic groups on contractible manifolds. Let (π,X,ρ) be such an action, where ρ:π→ Diff(X) is a homomorphism. We assume that ? extends to a smooth action of a real algebraic group G containing π. If such π has a nontrivial radical (i.e., unique maximal normal solvable subgroup), then we can apply the method of Seifert construction [14],[17] to yield that the quotient π\X supports the structure of an injective Seifert fibering with typical (resp. exceptional) fiber diffeomorphic to a solv (resp. infrasolv)-manifold (when π acts freely). When G is an amenable algebraic group, we can say about a uniqueness property for such actions. Namely, let (π_i, X_i, ρ_i) be actions as above (i= 1,2). Then, given an isomorphism f of π₁ onto ?₂, there is a diffeomorphism h: X₁→X₂ such that h(ρ₁(r)x)=ρ₂(f(r)h(x).As an application, we try to decide the structure of affine motions of some euclidean space $\text{R}$ⁿ. First we verify the conjecture of [17, 4 5], i.e., a compact complete affinely flat manifold admits a maximal toral action if its fundamental group has a nontrivial center. Second, a compact complete affinity flat manifold whose fundamental group is virtually polycyclic supports the structure of an infrasolvmanifold. This structure varies depending on its solvable kernel (if it is abelian or nilpotent, it must be a euclidean space form or an infranilmanifold respectively). If a group of the affine group A(n) acts properly discontinuously and with compact quotient of $\text{R}$ⁿ, then it is called an affine crystallographic group. Finally, we can say so far as to a uniqueness property that two virtually polycyclic affine crystallographic groups are conjugate inside Diff($\text{R}$ⁿ) if they are isomorphic (cf.[8]).     

The rate of convergence in Orey's theorem for Harris recurrent Markov chains with applications to renewal theory

We derive sufficient conditions for ∝ λ (dx)6Pⁿ(x, ·) - π6 to be of order o(ψ(n)^-1), where Pⁿ (x, A) are the transition probabilities of an aperiodic Harris recurrent Markov chain, π is the invariant probability measure, λ an initial distribution and ψ belongs to a suitable class of non-decreasing sequences. The basic condition involved is the ergodicity of order ψ, which in a countable state space is equivalent to $\text{Σ ψ(n)}\text{P}{}_{\mathrm{i}}\text{{τ}{}_{\mathrm{i}}\text{?n} <∞}$ for some i, where τ_i is the hitting time of the tate i. We also show that for a general Markov chain to be ergodic of order ψ it suffices that a corresponding condition is satisfied by a small set.We apply these results to non-singular renewal measures on $\text{R}$ providing a probabilisite method to estimate the right tail of the renewal measure when the increment distribution F satisfies ∝ tF(dt) 0; > 0 and ∝ ψ(t)(1- F(t))dt< ∞.     

A note on shadowing with chain transitivity

Let f : X → X be a continuous map of a compact metric space X. The map f induces in a natural way a map f_M on the space M(X) of probability measures on X, and a transformation f_K on the space K(X) of closed subsets of X. In this paper, we show that if (X, f) is a chain transitive system with shadowing property, then exactly one of the following two statements holds:

(a)

fⁿ and (f_K)ⁿ are syndetically sensitive for all n ? 1.

(b)

fⁿ and (f_K)ⁿ are equicontinuous for all n ? 1.

In particular, we show that for a continuous map f : X → X of a compact metric space X with infinite elements, if f is a chain transitive map with the shadowing property, then fⁿ and (f_K)ⁿ are syndetically sensitive for all n ? 1. Also, we show that if f_M (resp. f_K) is chain transitive and syndetically sensitive, and f_M (resp. f_K) has the shadowing property, then f is sensitive.In addition, we introduce the notion of ergodical sensitivity and present a sufficient condition for a chain transitive system (X, f) (resp. (M(X), f_M)) to be ergodically sensitive. As an application, we show that for a L-hyperbolic homeomorphism f of a compact metric space X, if f has the AASP, then fⁿ is syndetically sensitive and multi-sensitive for all n ? 1.     

The law of the Euler scheme for stochastic differential equations

Summary We study the approximation problem ofE f(X _T ) byE f(X _T ⁿ ), where (X _t ) is the solution of a stochastic differential equation, (X _T ⁿ ) is defined by the Euler discretization scheme with stepT/n, andf is a given function. For smoothf's, Talay and Tubaro have shown that the errorE f(X _T ) –f(X _T ⁿ ) can be expanded in powers of 1/n, which permits to construct Romberg extrapolation precedures to accelerate the convergence rate. Here, we prove that the expansion exists also whenf is only supposed measurable and bounded, under an additional nondegeneracy condition of Hörmander type for the infinitesimal generator of (X _t ): to obtain this result, we use the stochastic variations calculus. In the second part of this work, we will consider the density of the law ofX _T ⁿ and compare it to the density of the law ofX _T .     

Estimation de la transition de probabilité d'une chaîne de Markov doëblin-récurrente. étude du cas du processus autorégressif général d'ordre 1

We consider an homogenous Markov chain {X_n}. We estimate its transition probability density with kernel estimators. We apply these methods to the estimation of the unknown function f of the process defined by X₁ and X_n+1 = f(X_n) + ε_n, where {ε_n} is a noise (sequence of independent identically distributed random variables) of unknown law. The mean quadratic integrated rates of convergence are identical to those of classical density estimations. These risks are used here because we want some global informations about our estimates. We also study the average of those risks when the variance changes; it is shown that they reach a minimal value for some optimal variance. We study uniform convergence of our estimators. We finally estimate the variance of the noise and its density.     

Krengel-Lin decomposition for probability measures on hypergroups

A Markov operator P on a σ-finite measure space (X,Σ,m) with invariant measure m is said to have Krengel-Lin decomposition if L²(X)=E₀⊕L²(X,Σ_d) where E₀={f∈L²(X)∣‖Pⁿ(f)‖→0} and Σ_d is the deterministic σ-field of P. We consider convolution operators and we show that a measure λ on a hypergroup has Krengel-Lin decomposition if and only if the sequence converges to an idempotent or λ is scattered. We verify this condition for probabilities on Tortrat groups, on commutative hypergroups and on central hypergroups. We give a counter-example to show that the decomposition is not true for measures on discrete hypergroups.     

A central limit theorem for the sojourn times of strongly ergodic Markov chains

Let X_n be an irreducible aperiodic recurrent Markov chain with countable state space I and with the mean recurrence times having second moments. There is proved a global central limit theorem for the properly normalized sojourn times. More precisely, if $\text{t}{}^{\mathrm{(n}}\text{)}{}_{\mathrm{i}}\text{=Σ}{}^{\mathrm{n}}{}_{\mathrm{k=1}}\text{i}\text{?}{}_{\mathrm{i}}\text{(X}{}_{\mathrm{k}}\text{)}$, then the probability measures induced by {t⁽ⁿ⁾_i/√n?√nπ_i}_i?I(π_i being the ergotic distribution) on the Hilbert-space of square summable I-sequences converge weakly in this space to a Gaussian measure determined by a certain weak potential operator.     

On the cardinality of power homogeneous compacta

We prove that if X is a power homogeneous compact space then |X|⩽2^{c(X)·π_χ(X)}. This generalizes similar results of Arhangel'skiı̆, van Douwen and Ismail. We apply this result to get new estimates for the cardinality of (power) homogeneous compacta satisfying some special conditions.     

Devaney’s chaos on uniform limit maps

Let (X, d) be a compact metric space and f_n : X → X a sequence of continuous maps such that (f_n) converges uniformly to a map f. The purpose of this paper is to study the Devaney’s chaos on the uniform limit f. On the one hand, we show that f is not necessarily transitive even if all f_n mixing, and the sensitive dependence on initial conditions may not been inherited to f even if the iterates of the sequence have some uniform convergence, which correct two wrong claims in [1]. On the other hand, we give some equivalence conditions for the uniform limit f to be transitive and to have sensitive dependence on initial conditions. Moreover, we present an example to show that a non-transitive sequence may converge uniformly to a transitive map.     

Equiconvergence and equisummability of nonharmonic Fourier expansions with ordinary trigonometric series

Givenf ε L(?π, π), we consider its nonharmonic Fourier series \(f(x) \sim \sum c_n e^{i\lambda _n x} \) , where λ_n are the roots of the entire function L(z) = ∫ _-π ^π e ^izt dσ (t). We show that this series is equiconvergent, uniformly inside (-π, π), and equisummable with the Fourier series off with respect to the trigonometric system if σ′ (t) =k (t) (π - ∣t∣)^-α, α ε (0, 1), vark <∞, k (π ?0) ≠ 0,k (? π + 0) ≠ 0.     

Realcompactness of hyperspaces and extensions of open-closed maps

In response to questions of Ginsburg [9, 10], we prove that if cf(c)>ω₁, then there exists an open-closed, continuous map f from a normal, realcompact space X onto a space Y which is not realcompact. By his result the hyperspace 2^x of closed subsets of X is then not realcompact, and the extension μf(vf) of f to the topological completion (the Hewitt realcompactification) of X is not onto. The latter fact solves problems raised by Morita [16] and by Isiwata [12] both negatively. We also consider the problem whether or not the hyperspace of a hereditarily Lindelöf space is hereditarily realcompact.     

On the distribution of the singular values of Toeplitz matrices

In 1920, G. Szegö proved a basic result concerning the distribution of the eigenvalues {λ⁽ⁿ⁾_j} of the Toeplitz sections T_n [f], where f(Θ)∈L_∞( -π, π) is a real-valued function. Simple examples show that this result cannot hold in the case where f(Θ) is not real valued. In this note, we give an extension of this theorem for the singular values of T_n[f] when f(Θ)=f₀(Θ)R₀(Θ) with f₀(Θ) real-valued and R₀(Θ) continuous, periodic (with period 2π) and such that |R₀(Θ)|=1. In addition, we apply the basic theorem of Szegö to resolve a question of C. Moler.     

Asymptotic Optimality of Isoperimetric Constants

In this paper, we investigate the existence of L ²(π)-spectral gaps for π-irreducible, positive recurrent Markov chains with a general state space Ω. We obtain necessary and sufficient conditions for the existence of L ²(π)-spectral gaps in terms of a sequence of isoperimetric constants. For reversible Markov chains, it turns out that the spectral gap can be understood in terms of convergence of an induced probability flow to the uniform flow. These results are used to recover classical results concerning uniform ergodicity and the spectral gap property as well as other new results. As an application of our result, we present a rather short proof for the fact that geometric ergodicity implies the spectral gap property. Moreover, the main result of this paper suggests that sharp upper bounds for the spectral gap should be expected when evaluating the isoperimetric flow for certain sets. We provide several examples where the obtained upper bounds are exact.     

On the prediction of vector-valued random fields and the spectral distribution of their evanescent component

Let (X_m,n)_(m,n)∈Z² be a C^p-valued wide sense stationary process. We study the prediction theory of such processes according to different total orders on Z². In the case of a “rational order”, we give the spectral distribution of the resulting evanescent component and prove that for two different rational orders, the resulting evanescent components are mutually orthogonal.     

The π-π-theorem for manifold pairs with boundaries

The surgery obstruction of a normal map to a simple Poincaré pair (X, Y) lies in the relative surgery obstruction group L _*(π ₁(Y) → π ₁(X)). A well-known result of Wall, the so-called π-π-theorem, states that in higher dimensions a normal map of a manifold with boundary to a simple Poincaré pair with π ₁(X) ? π ₁(Y) is normally bordant to a simple homotopy equivalence of pairs. In order to study normal maps to a manifold with a submanifold, Wall introduced the surgery obstruction groups LP _* for manifold pairs and splitting obstruction groups LS _*. In the present paper, we formulate and prove for manifold pairs with boundary results similar to the π-π-theorem. We give direct geometric proofs, which are based on the original statements of Wall’s results and apply obtained results to investigate surgery on filtered manifolds.     

Convergence parameter associated with a Markov chain and a family of functions

The proposed definition of convergence parameter R(W) corresponding to a Markov chain X with a measurable state space (E,?) and any nonempty setW of bounded below measurable functions f: E → ? is wider than the well-known definition of convergence parameter R in the sense of Tweedie or Nummelin. Very often, R(W) < ∞, and there exists a set playing the role of the absorbing set inNummelin’s definition ofR. Special attention is paid to the case in whichE is locally compact, X is a Feller chain on E, and W coincides with the family ? ₀ ⁺ of all compactly supported continuous functions f ≥ 0 (f ? 0). In particular, certain conditions for R(? ₀ ⁺ )^?1 to coincide with the norm of an appropriate modification of the chain transition operator are found.     

On Monte Carlo methods for Bayesian multivariate regression models with heavy-tailed errors

We consider Bayesian analysis of data from multivariate linear regression models whose errors have a distribution that is a scale mixture of normals. Such models are used to analyze data on financial returns, which are notoriously heavy-tailed. Let π denote the intractable posterior density that results when this regression model is combined with the standard non-informative prior on the unknown regression coefficients and scale matrix of the errors. Roughly speaking, the posterior is proper if and only if n≥d+k, where n is the sample size, d is the dimension of the response, and k is number of covariates. We provide a method of making exact draws from π in the special case where n=d+k, and we study Markov chain Monte Carlo (MCMC) algorithms that can be used to explore π when n>d+k. In particular, we show how the Haar PX-DA technology studied in Hobert and Marchev (2008) [11] can be used to improve upon Liu’s (1996) [7] data augmentation (DA) algorithm. Indeed, the new algorithm that we introduce is theoretically superior to the DA algorithm, yet equivalent to DA in terms of computational complexity. Moreover, we analyze the convergence rates of these MCMC algorithms in the important special case where the regression errors have a Student’s t distribution. We prove that, under conditions on n, d, k, and the degrees of freedom of the t distribution, both algorithms converge at a geometric rate. These convergence rate results are important from a practical standpoint because geometric ergodicity guarantees the existence of central limit theorems which are essential for the calculation of valid asymptotic standard errors for MCMC based estimates.     

Absolute regularity and functions of Markov chains

We first give an extension of a theorem of Volkonskii and Rozanov characterizing the strictly stationary random sequences satisfying ‘absolute regularity’. Then a strictly stationary sequence {X_k, k = …, ?1, 0, 1,…} is constructed which is a 0?1 instantaneous function of an aperiodic Markov chain with countable irreducible state space, such that n^?2 var (X₁ + ? + X_n) approaches 0 arbitrarily slowly as n → ∞ and (X₁ + ? + X_n) is partially attracted to every infinitely divisible law.     

On almost sure convergence of amarts and martingales without the Radon-Nikodym property

It is shown here that for any Banach spaceE-valued amart (X _n) of classB, almost sure convergence off(X_n) tof(X) for eachf in a total subset ofE ^* implies scalar convergence toX.