Classical Moment Problem Research Articles

Construction of Solutions of Toda Lattices by the Classical Moment Problem

Construction of Solutions of Toda Lattices by the Classical Moment Problem

Projective Limit Techniques for the Infinite Dimensional Moment Problem

We deal with the following general version of the classical moment problem: when can a linear functional on a unital commutative real algebra A be represented as an integral with respect to a Radon measure on the character space X(A) of A equipped with the Borel sigma -algebra generated by the weak topology? We approach this problem by constructing X(A) as a projective limit of the character spaces of all finitely generated unital subalgebras of A. Using some fundamental results for measures on projective limits of measurable spaces, we determine a criterion for the existence of an integral representation of a linear functional on A with respect to a measure on the cylinder sigma -algebra on X(A) (resp. a Radon measure on the Borel sigma -algebra on X(A)) provided that for any finitely generated unital subalgebra of A the corresponding moment problem is solvable. We also investigate how to localize the support of representing measures for linear functionals on A. These results allow us to establish infinite dimensional analogues of the classical Riesz-Haviland and Nussbaum theorems as well as a representation theorem for linear functionals non-negative on a “partially Archimedean” quadratic module of A. Our results in particular apply to the case when A is the algebra of polynomials in infinitely many variables or the symmetric tensor algebra of a real infinite dimensional vector space, providing a unified setting which enables comparisons between some recent results for these instances of the moment problem.

On an application of the Boundary control method to classical moment problems.

We establish relationships between the classical moments problems which are problems of a construction of a measure supported on a real line, on a half-line or on an interval from prescribed set of moments with the Boundary control approach to a dynamic inverse problem for a dynamical system with discrete time associated with Jacobi matrices. We show that the solution of corresponding truncated moment problems is equivalent to solving some generalized spectral problems.

Numerical Algorithms for Estimating Probability Density Function Based on the Maximum Entropy Principle and Fup Basis Functions.

Estimation of the probability density function from the statistical power moments presents a challenging nonlinear numerical problem posed by unbalanced nonlinearities, numerical instability and a lack of convergence, especially for larger numbers of moments. Despite many numerical improvements over the past two decades, the classical moment problem of maximum entropy (MaxEnt) is still a very demanding numerical and statistical task. Among others, it was presented how Fup basis functions with compact support can significantly improve the convergence properties of the mentioned nonlinear algorithm, but still, there is a lot of obstacles to an efficient pdf solution in different applied examples. Therefore, besides the mentioned classical nonlinear Algorithm 1, in this paper, we present a linear approximation of the MaxEnt moment problem as Algorithm 2 using exponential Fup basis functions. Algorithm 2 solves the linear problem, satisfying only the proposed moments, using an optimal exponential tension parameter that maximizes Shannon entropy. Algorithm 2 is very efficient for larger numbers of moments and especially for skewed pdfs. Since both Algorithms have pros and cons, a hybrid strategy is proposed to combine their best approximation properties.

Modeling time‐varying higher‐order conditional moments: A survey

AbstractSince the introduction of the Autoregressive Conditional Heteroscedasticity (ARCH) model, the literature on modeling the time‐varying second‐order conditional moment has become increasingly popular in the last four decades. Its popularity is partly due to its success in capturing volatility in financial time series, which is useful for modeling and predicting risk for financial assets. A natural extension of this is to model time variation in higher‐order conditional moments, such as the third and fourth moments, which are related to skewness and kurtosis (tail risk). This leads to an emerging literature on time‐varying higher‐order conditional moments in the last two decades. This paper outlines recent developments in modeling time‐varying higher‐order conditional moments in the economics and finance literature.Using the Generalized Autoregressive Conditional Heteroscedasticity (GARCH) framework as a foundation, this paper provides an overview of the two most common approaches for modeling time‐varying higher‐order conditional moments: autoregressive conditional density (ARCD) and autoregressive conditional moment (ARCM). The discussion covers both the theoretical and empirical aspects of the literature. This includes the identification of the associated skewness–kurtosis domain by using the solutions to the classical moment problems, the structural and statistical properties of the models used to model the higher‐order conditional moments and the computational challenges in estimating these models. We also advocate the use of a maximum entropy density (MED) as an alternative method, which circumvents some of the issues prevalent in these common approaches.

A generalized trigonometric moment problem in a weighted L2(−∞,0) space

In this paper we consider a generalized trigonometric moment problem in a weighted L2(−∞,0) space. We obtain a solution which is in the closed span of some exponential system EΛ={tkeiλnt} in the weighted space. Our motivation is a classical trigonometric moment problem in L2(−π,π). Based on an entire function, a lower bound for the distance between exponential functions and the closed span of the remaining elements of the system EΛ is derived. An upper bound is then obtained for the norm of the elements of a biorthogonal system to EΛ. The proof of our result utilizes the notions of Bessel sequences and Riesz-Fischer sequences.

On the Moment Problem and Related Problems

Firstly, we recall the classical moment problem and some basic results related to it. By its formulation, this is an inverse problem: being given a sequence (yj)j∈ℕn of real numbers and a closed subset F⊆ℝn, n∈{1,2,…}, find a positive regular Borel measure μ on F such that ∫Ftjdμ=yj, j∈ℕn. This is the full moment problem. The existence, uniqueness, and construction of the unknown solution μ are the focus of attention. The numbers yj, j∈ℕn are called the moments of the measure μ. When a sandwich condition on the solution is required, we have a Markov moment problem. Secondly, we study the existence and uniqueness of the solutions to some full Markov moment problems. If the moments yj are self-adjoint operators, we have an operator-valued moment problem. Related results are the subject of attention. The truncated moment problem is also discussed, constituting the third aim of this work.

Measure-theoretic bounds on the spectral radius of graphs from walks

Let G be an undirected graph with adjacency matrix A and spectral radius ρ. Let wk,ϕk and ϕk(i) be, respectively, the number walks of length k, closed walks of length k and closed walks starting and ending at vertex i after k steps. In this paper, we propose a measure-theoretic framework which allows us to relate walks in a graph with its spectral properties. In particular, we show that wk,ϕk and ϕk(i) can be interpreted as the moments of three different measures, all of them supported on the spectrum of A. Building on this interpretation, we leverage results from the classical moment problem to formulate a hierarchy of new lower and upper bounds on ρ, as well as provide alternative proofs to several well-known bounds in the literature.

Moment Determinacy Versus q-moment Determinacy of Probability Distributions

Since the classical moment problem is an important issue deeply connected to various mathematical disciplines, its q-analogue based on the notion of q-moments has emerged in the study of q-distributions. For a wide class of probability distributions, both of these problems can be considered. The aim of this work is to establish a connection between the two moment problems. In this paper, the class $${\mathcal A}$$ of probability distributions possessing finite moments of all orders and support on $$(0, \infty )$$ is examined. For each $$q\in (0,1),$$ a distribution $$P\in {\mathcal A}$$ can be characterized with respect to moment-determinacy as well as q-moment determinacy. It is proved that the properties of P regarding these characterizations may differ, and that the q-moment determinacy of P may depend on the value of q.

On truncated and full classical Markov moment problems

On truncated and full classical Markov moment problems

Two-Dimensional Half-Strong Real Moment Problem and the Corresponding Block Matrices. Part I

We generalize the relationship between the classical moment problem and the spectral theory of Jacobi matrices. We present the solution of the two-dimensional half-strong moment problem and suggest an analog of Jacobi-type matrices associated with the two-dimensional half-strong moment problem and the corresponding system of polynomials orthogonal with respect to a measure with compact support in the real plane.

Двовимірна дійсна напівсильна проблема моментів та відповідні блочні матриці. I

УДК 517.9 Узагальнено зв’язок класичної проблеми моментiв iз спектральною теорiєю матриць Якобi. Наведено розв’язок двовимiрної напiвсильної проблеми моментiв та запропоновано аналог матриць типу Якобi, що вiдповiдає двовимiрнiй напiвсильнiй проблемi моментiв, та вiдповiдну систему полiномiв, ортогональних вiдносно мiри iз компактним носiєм на дiйснiй площинi.

Finite Toda lattice and classical moment problem

Finite Toda lattice and classical moment problem

Inverse problem for dynamical system associated with Jacobi matrices and classical moment problems

We consider Hamburger, Stieltjes and Hausdorff moment problems, that are problems of the determination of a Borel measure supported on the real axis, on the semi-axis or on the interval (0,1), from a prescribed set of moments. We propose a unified approach to these three problems based on the use of the auxiliary dynamical system with the discrete time associated with a semi-infinite Jacobi matrix. It is shown that the set of moments determines the inverse dynamic data for such a system. Using the ideas of the Boundary Control method, for every N∈N we can recover the spectral measure of N×N block of Jacobi matrix, which is a solution to a truncated moment problem. This problem is reduced to the finite-dimensional generalized spectral problem, whose matrices are constructed from moments and are connected with the well-known Hankel matrices by simple formulas. Thus the results on existence of solutions to Hamburger, Stieltjes and Hausdorff moment problems are naturally provided in terms of these matrices. We also obtain results on uniqueness of the solution of the moment problems, where as a main tool we use Krein-type equations of inverse problem.

Smallest eigenvalue of large Hankel matrices at critical point: Comparing conjecture with parallelised computation

We propose a novel parallel numerical algorithm for calculating the smallest eigenvalues of highly ill-conditioned Hankel matrices. It is based on the LDLT decomposition and involves finding a k × k sub-matrix of the inverse of the original N × N Hankel matrix HN−1. The computation involves extremely high precision arithmetic, message passing interface, and shared memory parallelisation. We demonstrate that this approach achieves good scalability on a high performance computing cluster (HPCC) which constitutes a major improvement of the earlier approaches. We use this method to study a family of Hankel matrices generated by the weight w(x)=e−xβ, supported on [0, ∞) and β > 0. Such weight generates a Hankel determinant, a fundamental object in random matrix theory. In the situation where β > 1/2, the smallest eigenvalue tends to 0 exponentially fast. If β < 1/2, which is the situation where the classical moment problem is indeterminate, then the smallest eigenvalue is bounded from below by a positive number. If β=1/2, it is conjectured that the smallest eigenvalue tends to 0 algebraically, with a precise exponent. The algorithm run on the HPCC producing a fantastic match between the theoretical value of 2/π and the numerical result.

Uniqueness of Banach space valued graphons

A Banach space valued graphon is a function W : ( Ω , A , π ) 2 → Z from a probability space to a Banach space with a separable predual, measurable in a suitable sense, and lying in appropriate L p -spaces. As such we may consider W ( x , y ) as a two-variable random element of the Banach space. A two-dimensional analogue of moments can be defined with the help of graphs and weak-* evaluations, and a natural question that then arises is whether these generalized moments determine the function W uniquely – up to measure preserving transformations. The main motivation comes from the theory of multigraph limits, where these graphons arise as the natural limit objects for convergence in a generalized homomorphism sense. Our main result is that this holds true under some Carleman-type condition, but fails in general even with Z = R , for reasons related to the classical moment-problem. In particular, limits of multigraph sequences are uniquely determined – up to measure preserving transformations – whenever the tails of the edge-distributions stay small enough.

Inverse of Infinite Hankel Moment Matrices

Let $(s_n)_{n\ge 0}$ denote an indeterminate Hamburger moment sequence and let $\mathcal H=\{s_{m+n}\}$ be the corresponding positive definite Hankel matrix. We consider the question if there exists an infinite symmetric matrix $\mathcal A=\{a_{j,k}\}$, which is an inverse of $\mathcal H$ in the sense that the matrix product $\mathcal A\mathcal H$ is defined by absolutely convergent series and $\mathcal A\mathcal H$ equals the identity matrix $\mathcal I$, a property called (aci). A candidate for $\mathcal A$ is the coefficient matrix of the reproducing kernel of the moment problem, considered as an entire function of two complex variables. We say that the moment problem has property (aci), if (aci) holds for this matrix $\mathcal A$. We show that this is true for many classical indeterminate moment problems but not for the symmetrized version of a cubic birth-and-death process studied by Valent and co-authors. We consider mainly symmetric indeterminate moment problems and give a number of sufficient conditions for (aci) to hold in terms of the recurrence coefficients for the orthonormal polynomials. A sufficient condition is a rapid increase of the recurrence coefficients in the sense that the quotient between consecutive terms is uniformly bounded by a constant strictly smaller than one. We also give a simple example, where (aci) holds, but an inverse matrix of $\mathcal H$ is highly non-unique.

The Euler and Springer numbers as moment sequences

I study the sequences of Euler and Springer numbers from the point of view of the classical moment problem.

The Classical Moment Problem and Generalized Indefinite Strings

We show that the classical Hamburger moment problem can be included in the spectral theory of generalized indefinite strings. More precisely, we introduce the class of Krein–Langer strings and show that there is a bijective correspondence between moment sequences and this class of generalized indefinite strings. This result can be viewed as a complement to the classical results of Krein on the connection between the Stieltjes moment problem and Krein–Stieltjes strings and Kac on the connection between the Hamburger moment problem and 2times 2 canonical systems with Hamburger Hamiltonians.

Linear Matrix Inequalities for Physically Consistent Inertial Parameter Identification: A Statistical Perspective on the Mass Distribution

With the increased application of model-based whole-body control in legged robots, there has been a resurgence of research interest into methods for accurate system identification. An important class of methods focuses on the inertial parameters of rigid-body systems. These parameters consist of the mass, first mass moment (related to center of mass location), and rotational inertia matrix of each link. The main contribution of this paper is to formulate physical-consistency constraints on these parameters as Linear Matrix Inequalities (LMIs). The use of these constraints in identification can accelerate convergence and increase robustness to noisy data. It is critically observed that the proposed LMIs are expressed in terms of the covariance of the mass distribution, rather than its rotational moments of inertia. With this perspective, connections to the classical problem of moments in mathematics are shown to yield new bounding-volume constraints on the mass distribution of each link. While previous work ensured physical plausibility or used convex optimization in identification, the LMIs here uniquely enable both advantages. Constraints are applied to identification of a leg for the MIT Cheetah 3 robot. Detailed properties of transmission components are identified alongside link inertias, with parameter optimization carried out to global optimality through semidefinite programming.