Measure Μ0 Research Articles

Some Results on the Free Poisson Distribution

Let K+(μi)={Qsiμi,si∈(m0μi,m+μi)}, i=1,2, be two CSK families generated by the nondegenerate probability measures μ1 and μ2 with support bounded from above. Define the set of measures L=K+(μ1)•K+(μ2)={Qs1μ1•Qs2μ2,s1∈(m0μ1,m+μ1)ands2∈(m0μ2,m+μ2)}, where Qs1μ1•Qs2μ2 denotes the Fermi convolution of Qs1μ1 and Qs2μ2. We prove that if L is still a CSK family (that is, L=K+(σ) for some nondegenerate probability measure ()σ), then the probability measures σ, μ1 and μ2 are of the free Poisson type and follow the free Poisson law up to affinity. The same result, regarding the free Poisson measure, is obtained if we consider the t-deformed free convolution t replacing the Fermi convolution • in the family of measures L.

Geometric mean of probability measures and geodesics of Fisher information metric

AbstractThe space of all probability measures having positive density function on a connected compact smooth manifold M, denoted by , carries the Fisher information metric G. We define the geometric mean of probability measures by the aid of which we investigate information geometry of , equipped with G. We show that a geodesic segment joining arbitrary probability measures μ1 and μ2 is expressed by using the normalized geometric mean of its endpoints. As an application, we show that any two points of can be joined by a unique geodesic. Moreover, we prove that the function ℓ defined by , , , gives the Riemannian distance function on . It is shown that geodesics are all minimal.

SuperMix: Sparse regularization for mixtures

This paper investigates the statistical estimation of a discrete mixing measure μ0 involved in a kernel mixture model. Using some recent advances in ℓ1-regularization over the space of measures, we introduce a “data fitting and regularization” convex program for estimating μ0 in a grid-less manner from a sample of mixture law, this method is referred to as Beurling-LASSO. Our contribution is two-fold: we derive a lower bound on the bandwidth of our data fitting term depending only on the support of μ0 and its so-called “minimum separation” to ensure quantitative support localization error bounds; and under a so-called “nondegenerate source condition” we derive a nonasymptotic support stability property. This latter shows that for a sufficiently large sample size n, our estimator has exactly as many weighted Dirac masses as the target μ0, converging in amplitude and localization towards the true ones. Finally, we also introduce some tractable algorithms for solving this convex program based on “Sliding Frank–Wolfe” or “Conic Particle Gradient Descent”. Statistical performances of this estimator are investigated designing a so-called “dual certificate”, which is appropriate to our setting. Some classical situations as, for example, mixtures of super-smooth distributions (see, e.g., Gaussian distributions) or ordinary-smooth distributions (see, e.g., Laplace distributions), are discussed at the end of the paper.

Perturbed finite-state Markov systems with holes and Perron complements of Ruelle operators

We consider a perturbed system ($$\Sigma_A^+$$,ϕ(ϵ, ·)) with topologically transitive subshift of finite type $$\Sigma_A^+$$ and Holder continuous functions ϕ(ϵ, ·) defined on $$\Sigma_A^+$$ endowed with small parameter ϵ > 0. Through our choice of ϕ(ϵ, ·), we realize the situation that the perturbed system has a unique Gibbs measure μϵ of the potential ϕ(ϵ, ·) for each ϵ > 0 and on the other hand the unperturbed system possesses several Gibbs measures μ1, μ2,…, μm of the limit potential at ϵ = 0. In this paper, we give a necessary and sufficient condition for convergence of the measure μϵ using the notion of Perron complements of Ruelle operators. Our results can be applied also to the problems of convergence of stationary distributions of perturbed Markov chains with holes.

Noise stability is computable and approximately low-dimensional

Questions of noise stability play an important role in hardness of approximation in computer science as well as in the theory of voting. In many applications, the goal is to find an optimizer of noise stability among all possible partitions of Rn for n ≥ 1 to k parts with given Gaussian measures μ1, ..., μk. We call a partition e-optimal, if its noise stability is optimal up to an additive e. In this paper, we give an explicit, computable function n(e) such that an e-optimal partition exists in Rn(e). This result has implications for the computability of certain problems in non-interactive simulation, which are addressed in a subsequent work.

Extension of measures on two disjoint rings

We present a method of starting with two measures μ1 and μ2 defined on disjoint rings R1 and R2 respectively, and defining another measure which is an extension of both μ1 and μ2. The method is a generalization of a classical method of extending a measure. An outer measure μ* is defined on a hereditary σ – ring H(R1, R2) which is a superset of both R1 and R2, and then a complete measure on the class of μ*-measurable sets that belong to H(R1, R2) is defined in terms of μ*.

Perturbation of chains of de Branges spaces

We investigate the structure of the set of de Branges spaces of entire functions which are contained in a space L2(μ). Thereby, we follow a perturbation approach. The main result is a growth dependent stability theorem. Namely, assume that measures μ1 and μ2 are close to each other in a sense quantified relative to a proximate order. Consider the sections of corresponding chains of de Branges spaces C1 and C2 which consist of those spaces whose elements have finite (possibly zero) type with respect to the given proximate order. Then either these sections coincide or one is smaller than the other but its complement consists of only a (finite or infinite) sequence of spaces. Among other situations, we apply—and refine—this general theorem in two important particular situations In the proof of the main result, we employ a method used by P. Yuditskii in the context of density of polynomials. Another vital tool is the notion of the index of a chain, which is a generalisation of the index of determinacy of a measure having all power moments. We undertake a systematic study of this index, which is also of interest on its own right.

Absolute Continuity of Wasserstein Barycenters Over Alexandrov Spaces

AbstractIn this paper, we prove that on a compact, n-dimensional Alexandrov space with curvature at least −1, the Wasserstein barycenter of Borel probability measures μ1 ,… , μm is absolutely continuous with respect to the n-dimensional Hausdorff measure if one of them is.

Optimal transportation for generalized Lagrangian

This paper deals with the optimal transportation for generalized Lagrangian L = L(x, u, t), and considers the following cost function: $$c\left( {x,y} \right) = \mathop {\inf }\limits_{\begin{array}{*{20}{c}} {x\left( 0 \right) = x} {x\left( 1 \right) = y} {u \in U} \end{array}} \int_0^1 {L\left( {x\left( s \right),u\left( {x\left( s \right),s} \right),s} \right)ds} ,$$ where U is a control set, and x satisfies the ordinary equation $$\dot x\left( s \right) = f\left( {x\left( s \right),u\left( {x\left( s \right),s} \right)} \right).$$ It is proved that under the condition that the initial measure μ 0 is absolutely continuous w.r.t. the Lebesgue measure, the Monge problem has a solution, and the optimal transport map just walks along the characteristic curves of the corresponding Hamilton-Jacobi equation: $$\left\{ {_{V\left( {0,x} \right) = {\phi _0}\left( x \right).}^{{V_t}\left( {t,x} \right) + \mathop {\sup }\limits_{u \in U} \left\langle {{V_x}\left( {t,x} \right),f\left( {x,u\left( {x\left( t \right),t} \right),t} \right) - L\left( {x\left( t \right),u\left( {x\left( t \right),t} \right),t} \right)} \right\rangle = 0,}} \right.$$

Multiple Spectra of Bernoulli Convolutions

AbstractLet μλ be the Bernoulli convolution associated with λ ∈ (0, 1). The well-known result of Jorgensen and Pedersen shows that if λ = 1/(2k) for some k ∈ ℕ, then μ1/(2k) is a spectral measure with spectrum Γ(1/(2k)). The recent research on the spectrality of μλ shows that μλ is a spectral measure only if λ = 1/(2k) for some k ∈ ℕ. Moreover, for certain odd integer p, the multiple set pΓ(1/(2k)) is also a spectrum for μ1/(2k). This is surprising because some spectra for the measure μ1/(2k) are thinning. In this paper we mainly characterize the number p that has the above property. By applying the properties of congruences and the order of elements in the finite group, we obtain several conditions on p such that pΓ(1/(2k)) is a spectrum for μ1/(2k).

On the Mahler measure of the Coxeter polynomial of an algebra

Let A be a finite dimensional algebra over an algebraically closed field k. Assume A is a basic connected and triangular algebra with n pairwise non-isomorphic simple modules. We consider the Coxeter transformationϕA(T) as the automorphism of the Grothendieck group K0(A) induced by the Auslander–Reiten translation τ in the derived category Derb(modA) of the module category modA of finite dimensional left A-modules. We say that A is of cyclotomic type if the characteristic polynomial χA of ϕA is a product of cyclotomic polynomials, equivalently, if the Mahler measureM(χA)=1. In [6] we have considered many examples of algebras of cyclotomic type in the representation theory literature. In this paper we study the Mahler measure of the Coxeter polynomial of accessible algebras. In 1933, D.H. Lehmer found that the polynomial T10+T9−T7−T6−T5−T4−T3+T+1 has Mahler measure μ0=1.176280..., and he asked whether there exist any smaller values exceeding 1. In this paper we prove that for any accessible algebra A either M(χA)=1 or M(χB)≥μ0 for some convex subcategory B of A. We introduce interlaced tower of algebrasAm,…,An with m≤n−2 satisfyingχAs+1=(T+1)χAs−TχAs−1 for m+1≤s≤n−1. We prove that, if SpecϕAn⊂S1∪R+ and An is not of cyclotomic type, then M(χAm)<M(χAn). We construct several examples.

On the Behaviour of the spectral characteristic of Feigenbaum’s map

Let Tg: [−1, 1] → [−1, 1] be the Feigenbaum map. It is well known that Tg has a Cantor-type attractor F and a unique invariant measure µ0 supported on F. The corresponding unitary operator (Ugφ)(x) = φ(g(x)) has pure point spectrum consisting of eigenvalues λn,r, n ≥ 1, 0 ≤ r ≤ 2n−1 − 1 with eigenfunctions er(n)(x). Suppose that f ∈ C1([−1, 1]), f′ is absolutely continuous on [−1, 1] and f″ ∈ Lp([−1, 1], dµ0), p > 1. Consider the sum of the amplitudes of the spectral measure of f: $$ Sn(f): = \sum\limits_{r = 0}^{2^n - 1} {|\rho _r^{(n)} |^2 ,\rho _r^{(n)} = \int\limits_{ - 1}^1 {f(x)\overline {e_r^{(n)} (x)} d\mu _o } } (x). $$ Using the thermodynamic formalism for Tg we prove that Sn(f) ∼ 2−nqn, as n → ∞, where the constant q ∈ (0, 1) does not depend on f.

Orthogonal polynomials in the normal matrix model with a cubic potential

We consider the normal matrix model with a cubic potential. The model is ill-defined, and in order to regularize it, Elbau and Felder introduced a model with a cut-off and corresponding system of orthogonal polynomials with respect to a varying exponential weight on the cut-off region on the complex plane. In the present paper we show how to define orthogonal polynomials on a specially chosen system of infinite contours on the complex plane, without any cut-off, which satisfy the same recurrence algebraic identity that is asymptotically valid for the orthogonal polynomials of Elbau and Felder.The main goal of this paper is to develop the Riemann–Hilbert (RH) approach to the orthogonal polynomials under consideration and to obtain their asymptotic behavior on the complex plane as the degree n of the polynomial goes to infinity. As the first step in the RH approach, we introduce an auxiliary vector equilibrium problem for a pair of measures (μ1,μ2) on the complex plane. We then formulate a 3×3 matrix valued RH problem for the orthogonal polynomials in hand, and we apply the nonlinear steepest descent method of Deift–Zhou to the asymptotic analysis of the RH problem. The central steps in our study are a sequence of transformations of the RH problem, based on the equilibrium vector measure (μ1,μ2), and the construction of a global parametrix.The main result of this paper is a derivation of the large n asymptotics of the orthogonal polynomials on the whole complex plane. We prove that the distribution of zeros of the orthogonal polynomials converges to the measure μ1, the first component of the equilibrium measure. We also obtain analytical results for the measure μ1 relating it to the distribution of eigenvalues in the normal matrix model which is uniform in a domain bounded by a simple closed curve.

Inference in ϕ-families of distributions

This paper is devoted to the study of the parametric family of multivariate distributions obtained by minimizing a convex functional under linear constraints. Under certain assumptions on the convex functional, it is established that this family admits an affine parameterization, and parametric estimation from an i.i.d. random sample is studied. It is also shown that the members of this family are the limit distributions arising in inference based on empirical likelihood. As a consequence, given a probability measure μ0 and an i.i.d. random sample drawn from μ0, nonparametric confidence domains on the generalized moments of μ0 are obtained.

Uniform measures on inverse limit spaces

Motivated by problems from dynamic economic models, we consider the problem of defining a uniform measure on inverse limit spaces. Let where X is a compact metric space and f is continuous, onto and piecewise one-to-one and . Then starting with a measure μ1 on the Borel sets , we recursively construct a sequence of probability measures on satisfying for each and . This sequence of probability measures is then uniquely extended to a probability measure on the inverse limit space Y. If μ1 is a uniform measure, we argue that the measure induced on the inverse limit space by the recursively constructed sequence of measures is a uniform measure. As such, the measure has uses in economic theory for policy evaluation and in dynamical systems in providing an ambient measure (when Lebesgue measure is not available) with which to define a Sinai–Ruelle–Bowen (SRB) measure or a metric attractor for the shift map on the inverse limit space.

Product of uniform measures

For i = (1, 2), let Xi be Hausdorff uniform spaces and μi uniform measures on Xi. We determine the existence of the product uniform measure μ1⊗μ2 on X1 × X2 and prove a Fubini type theorem and a continuity property. The result is extended to vector-valued uniform measures.

Minimal topological actions do not determine the measurable orbit equivalence class

We construct an amenable action Φ of a non-amenable group Γ on a discrete space. This action extends to a minimal topological action Φ of Γ on a Cantor set C. We show that Φ is non-uniquely ergodic and furthermore there exist ergodic invariant measures μ1 and μ2 such that (Φ,C,μ1) and (Φ,C,μ2) are not orbit equivalent measurable equivalence relations. This also provides an instance of the failure of equivalence between the notions of “global” and “local” amenability for countable equivalence relations.

Hamiltonian ODEs in the Wasserstein space of probability measures

AbstractIn this paper we consider a Hamiltonian H on 𝒫2(ℝ2d), the set of probability measures with finite quadratic moments on the phase space ℝ2d = ℝd × ℝd, which is a metric space when endowed with the Wasserstein distance W2. We study the initial value problem dμt/dt + ∇ · (𝕁dvtμt) = 0, where 𝕁d is the canonical symplectic matrix, μ0 is prescribed, and vt is a tangent vector to 𝒫2(ℝ2d) at μt, belonging to ∂H(μt), the subdifferential of H at μt. Two methods for constructing solutions of the evolutive system are provided. The first one concerns only the case where μ0 is absolutely continuous. It ensures that μt remains absolutely continuous and vt = ∇H(μt) is the element of minimal norm in ∂H(μt). The second method handles any initial measure μ0. If we further assume that H is λ‐convex, proper, and lower‐semicontinuous on 𝒫2(ℝ2d), we prove that the Hamiltonian is preserved along any solution of our evolutive system, H(μt) = H(μ0). © 2007 Wiley Periodicals, Inc.

Feynman’s Operational Calculi: Spectral Theory for Noncommuting Self-adjoint Operators

The spectral theorem for commuting self-adjoint operators along with the associated functional (or operational) calculus is among the most useful and beautiful results of analysis. It is well known that forming a functional calculus for noncommuting self-adjoint operators is far more problematic. The central result of this paper establishes a rich functional calculus for any finite number of noncommuting (i.e. not necessarily commuting) bounded, self-adjoint operators A1,..., An and associated continuous Borel probability measures μ1, ⋯, μn on [0,1]. Fix A1,..., An. Then each choice of an n-tuple \(\left(\mu_1,\ldots,\mu_n\right)\) of measures determines one of Feynman’s operational calculi acting on a certain Banach algebra of analytic functions even when A1, ..., An are just bounded linear operators on a Banach space. The Hilbert space setting along with self-adjointness allows us to extend the operational calculi well beyond the analytic functions. Using results and ideas drawn largely from the proof of our main theorem, we also establish a family of Trotter product type formulas suitable for Feynman’s operational calculi.

Uniqueness of measure extensions in Banach spaces

Let X be a Banach space, B ⊂ BX∗ a norming set and T(X,B) the topology on X of pointwise convergence on B. In this paper we study the following question: given two (non negative, countably additive and finite) measures μ1 and μ2 on Baire(X,w) which coincide on Baire(X,T(X,B)), does it follow that μ1 = μ2? It turns out that this is not true in general, although the answer is affirmative provided that both μ1 and μ2 are convexly τ -additive (e.g. when X has the Pettis Integral Property). For a Banach space Y not containing isomorphic copies of `1, we show that Y ∗ has the Pettis Integral Property if and only if every measure on Baire(Y ∗, w∗) admits a unique extension to Baire(Y ∗, w). We also discuss the coincidence of the two σ-algebras involved in such results. Some other applications are given.