Moment convergence in regularized estimation under multiple and mixed-rates asymptotics

In M-estimation under standard asymptotics, the weak convergence combined with the polynomial type large deviation estimate of the associated statistical random field Yoshida (2011) provides us with not only the asymptotic distribution of the associated M-estimator but also the convergence of its moments, the latter playing an important role in theoretical statistics. In this paper, we study the above program for statistical random fields of multiple and also possibly mixedrates type in the sense of Radchenko (2008) where the associated statistical random fields may be nondifferentiable and may fail to be locally asymptotically quadratic. Consequently, a very strong mode of convergence of a wide range of regularized M-estimators is ensured.Our results are applied to regularized estimation of an ergodic diffusion observed at high frequency.     

First order absolute moment of Meyer-König and Zeller operators and their approximation for some absolutely continuous functions

A sharp estimate is given for the first order absolute moment of Meyer-König and Zeller operators M _n . This estimate is then used to prove convergence of approximation of a class of absolutely continuous functions by the operators M _n . The condition considered here is weaker than the condition considered in a previous paper and the rate of convergence we obtain is asymptotically the best possible.     

Estimates of reachable sets of control systems with nonlinearity and parametric perturbations

In the Banach space L₁(M, τ) of operators integrable with respect to a tracial state τ on a von Neumann algebra M, convergence is analyzed. A notion of dispersion of operators in L₂(M, τ) is introduced, and its main properties are established. A convergence criterion in L₂(M, τ) in terms of the dispersion is proposed. It is shown that the following conditions for X ∈ L₁(M, τ) are equivalent: (i) τ(X) = 0, and (ii) ‖I + zX‖₁ ≥ 1 for all z ∈ C. A.R. Padmanabhan’s result (1979) on a property of the norm of the space L₁(M, τ) is complemented. The convergence in L₂(M, τ) of the imaginary components of some bounded sequences of operators from M is established. Corollaries on the convergence of dispersions are obtained.     

Finite Groups Whose <Emphasis Type="Italic">n</Emphasis>-Maximal Subgroups Are Modular

Let G be a finite group. If M_n< M_n?1< · · · < M₁< M₀ = G with Mi a maximal subgroup of M_i?1 for all i = 1,..., n, then M_n (n > 0) is an n-maximal subgroup of G. A subgroup M of G is called modular provided that (i) 〈X,M ∩ Z〉 = 〈X,M〉 ∩ Z for all X ≤ G and Z ≤ G such that X ≤ Z, and (ii) 〈M,Y ∩ Z〉 = 〈M,Y 〉 ∩ Z for all Y ≤ G and Z ≤ G such that M ≤ Z. In this paper, we study finite groups whose n-maximal subgroups are modular.     

An Algorithm for Approximating the Second Moment of the Normalizing Constant Estimate from a Particle Filter

We propose a new algorithm for approximating the non-asymptotic second moment of the marginal likelihood estimate, or normalizing constant, provided by a particle filter. The computational cost of the new method is O(M) per time step, independently of the number of particles N in the particle filter, where M is a parameter controlling the quality of the approximation. This is in contrast to O(M N) for a simple averaging technique using M i.i.d. replicates of a particle filter with N particles. We establish that the approximation delivered by the new algorithm is unbiased, strongly consistent and, under standard regularity conditions, increasing M linearly with time is sufficient to prevent growth of the relative variance of the approximation, whereas for the simple averaging technique it can be necessary to increase M exponentially with time in order to achieve the same effect. This makes the new algorithm useful as part of strategies for estimating Monte Carlo variance. Numerical examples illustrate performance in the context of a stochastic Lotka–Volterra system and a simple AR(1) model.     

Joint functional convergence of partial sums and maxima for linear processes<Superscript>*</Superscript>

For linear processes with independent identically distributed innovations that are regularly varying with tail index α ∈ (0, 2), we study the functional convergence of the joint partial-sum and partial-maxima processes. We derive a functional limit theorem under certain assumptions on the coefficients of the linear processes, which enable the functional convergence in the space of ?²-valued càdlàg functions on [0, 1] with the Skorokhod weak M₂ topology.We also obtain a joint convergence in the M₂ topology on the first coordinate and in theM₁ topology on the second coordinate.     

Symplectic geometry on moduli spaces of <Emphasis Type="Italic">J</Emphasis>-holomorphic curves

Let (M, ω) be a symplectic manifold, and Σ a compact Riemann surface. We define a 2-form \({\omega_{\mathcal{S}_{i}(\Sigma)}}\) on the space \({\mathcal{S}_{i}(\Sigma)}\) of immersed symplectic surfaces in M, and show that the form is closed and non-degenerate, up to reparametrizations. Then we give conditions on a compatible almost complex structure J on (M, ω) that ensure that the restriction of \({\omega_{\mathcal{S}_{i}(\Sigma)}}\) to the moduli space of simple immersed J-holomorphic Σ-curves in a homology class \({A \in {H}_2(M,\,\mathbb{Z})}\) is a symplectic form, and show applications and examples. In particular, we deduce sufficient conditions for the existence of J-holomorphic Σ-curves in a given homology class for a generic J.     

Strict <Emphasis Type="Italic">K</Emphasis>-monotonicity and <Emphasis Type="Italic">K</Emphasis>-order continuity in symmetric spaces

This paper is devoted to strict K-monotonicity and K-order continuity in symmetric spaces. Using a local approach to the geometric structure in a symmetric space E we investigate a connection between strict K-monotonicity and global convergence in measure of a sequence of the maximal functions. Next, we solve an essential problem whether an existence of a point of K-order continuity in a symmetric space E on \([0,\infty )\) implies that the embedding \(E\hookrightarrow {L^1}[0,\infty )\) does not hold. We present a complete characterization of an equivalent condition to K-order continuity in a symmetric space E using a notion of order continuity and the fundamental function of E. We also investigate a relationship between strict K-monotonicity and K-order continuity in symmetric spaces and show some examples of Lorentz spaces and Marcinkiewicz spaces having these properties or not. Finally, we discuss a local version of a crucial correspondence between order continuity and the Kadec–Klee property for global convergence in measure in a symmetric space E.     

Processes of <Emphasis Type="Italic">r</Emphasis><Superscript><Emphasis Type="Italic">t</Emphasis><Emphasis Type="Italic">h</Emphasis></Superscript> largest

For integers n ≥ r, we treat the rth largest of a sample of size n as an \(\mathbb {R}^{\infty }\)-valued stochastic process in r which we denote as M^(r). We show that the sequence regarded in this way satisfies the Markov property. We go on to study the asymptotic behavior of M^(r) as r → ∞, and, borrowing from classical extreme value theory, show that left-tail domain of attraction conditions on the underlying distribution of the sample guarantee weak limits for both the range of M^(r) and M^(r) itself, after norming and centering. In continuous time, an analogous process Y^(r) based on a two-dimensional Poisson process on \(\mathbb {R}_{+}\times \mathbb {R}\) is treated similarly, but we note that the continuous time problems have a distinctive additional feature: there are always infinitely many points below the rth highest point up to time t for any t >?0. This necessitates a different approach to the asymptotics in this case.     

Isometries with dense windings of the torus in <Emphasis Type="Italic">C</Emphasis>(<Emphasis Type="Italic">M</Emphasis>)

Let C(M) be the space of all continuous functions on M? ?. We consider the multiplication operator T: C(M) → C(M) defined by Tf(z) = zf(z) and the torus

$$O(M) = \left\{ {f:M \to \mathbb{C} \ntrianglelefteq \left\| f \right\| = \left\| {\frac{1}{f}} \right\| = 1} \right\}$$

. If M is a Kronecker set, then the T-orbits of the points of the torus ½O(M) are dense in ½O(M) and are ½-dense in the unit ball of C(M).

     

<Emphasis Type="Italic">M</Emphasis>-slenderness

Analogues of Nunke’s theorem are proved which characterize variants of slenderness. For a bounded monotone subgroup M of ? ^ω , a torsion-free reduced abelian group G is M-slender if, and only if, there is no monomorphism from M into G. It is consistent relative to ordinary set theory (ZFC) that if M ≠ ? ^ω is an unbounded monotone subgroup of ? ^ω , then a torsion-free reduced abelian group G is M-slender if, and only if, there is no monomorphism from M into G.     

Spectral Properties of <Emphasis Type="Italic">k</Emphasis>-Quasi-<Emphasis Type="Italic">M</Emphasis>-hyponormal Operators

In this paper, we study the k-quasi-M-hyponormal operator and mainly prove that if T is a k-quasi-M-hyponormal operator, then \(\sigma _{ja}(T)\backslash \{0\}=\sigma _{a}(T)\backslash \{0\}\), and the spectrum is continuous on the class of all k-quasi-M-hyponormal operators; let \(d_{AB}\in B(B(H))\) denote either the generalized derivation \(\delta _{AB}= L_{A}-R_{B}\) or the elementary operator \(\Delta _{AB} =L_{A}R_{B}- I\), we show that if A and \(B^{*}\) are k-quasi-M-hyponormal operators, then \(d_{AB}\) is polaroid and generalized Weyl’s theorem holds for \(f(d_{AB})\), where f is an analytic function on \(\sigma (d_{AB})\) and f is not constant on each connected component of the open set U containing \(\sigma (d_{AB})\). In additon, we discuss the hyperinvariant subspace problem for k-quasi-M-hyponormal operators.     

Deterministic sparse FFT for <Emphasis Type="Italic">M</Emphasis>-sparse vectors

In this paper, we derive a new deterministic sparse inverse fast Fourier transform (FFT) algorithm for the case that the resulting vector is sparse. The sparsity needs not to be known in advance but will be determined during the algorithm. If the vector to be reconstructed is M-sparse, then the complexity of the method is at most \(\mathcal {O} (M^{2} \log N)\) if M ² < N and falls back to the usual \(\mathcal {O}(N \log N) \) algorithm for M ² ≥ N. The method is based on the divide-and-conquer approach and may require the solution of a Vandermonde system of size at most M × M at each iteration step j if M ² < 2 ^j . To ensure the stability of the Vandermonde system, we propose to employ a suitably chosen parameter σ that determines the knots of the Vandermonde matrix on the unit circle.     

Regularity in the Local CR Embedding Problem

We consider a formally integrable, strictly pseudoconvex CR manifold M of hypersurface type, of dimension 2n?1≥7. Local CR, i.e., holomorphic, embeddings of M are known to exist from the works of Kuranishi and Akahori. We address the problem of regularity of the embedding in standard Hölder spaces C ^a (M), a∈R. If the structure of M is of class C ^m , m∈Z, 4≤m≤∞, we construct a local CR embedding near each point of M. This embedding is of class C ^a , for every a, 0≤a<m+(1/2). Our method is based on Henkin’s local homotopy formula for the embedded case, some very precise estimates for the solution operators in it, and a substantial modification of a previous Nash–Moser argument due to the second author.     

Theoretical basis of the iterative process of the joint assessment of difficulties in tasks and levels of training students

In this paper, we study the iterative process of the joint numerical assessment of levels of training students and difficulties in tasks of diagnostic tools using the dichotomous response matrix A of size N × M with allowance for the contribution of tasks of different difficulty to the assessments obtained. It is shown that not for any matrix A there exist infinite iterative sequences, and in the case of their existence, they do not always converge. A wide range of sufficient conditions for their convergence have been obtained, which are based on the following: (1) the matrix A contains at least three different columns; (2) if one places the columns of the matrix A in non-decreasing order of column sums, then for any position of the vertical dividing line between the columns there exists a row, which has at least one unity to the left of the line and at least one zero to the right of the line. It is established that the response matrix A obtained as a result of testing reliability satisfies these two conditions. The properties of such matrices have been studied. In particular, the equivalence of the above-mentioned conditions of primitiveness of the square matrix B of order M with the entries \({b_{ij}} = \sum\limits_{\ell = 1}^N {(1 - {a_{\ell i}}){a_{\ell i}}} \) has been proved. Using the matrix analysis, we have proved that the primitiveness of the matrix B ensures the convergence of iterative sequences, as well as independence of their limits of the choice of the initial approximation. We have estimated the rate of convergence of these sequences and found their limits.     

On <Emphasis Type="BoldItalic">S</Emphasis>-strong Mori modules

Let A be an integral domain, \(S\subseteq A\) be a multiplicative set and M a w-module as an A-module. In this paper we investigate S-SM-modules. We give an S-version of the result of Wang and McCasland (Commun Algebra 25:1285–1306, 1997) in the case where S is countable. We prove that M is an S-SM-module if and only if every increasing sequence of w-submodules of M is S-stationary if and only if every increasing sequence of S-w-finite w-submodules of M is S-stationary if and only if every increasing sequence of w-finite type submodules of M is S-stationary.     

The general rigidity result for bundles of <Emphasis Type="Italic">A</Emphasis>-covelocities and <Emphasis Type="Italic">A</Emphasis>-jets

Let M be an m-dimensional manifold and A = D _k ^r /I = R⊕N _A a Weil algebra of height r. We prove that any A-covelocity T _x ^A f ∈ T _x ^A *M, x ∈ M is determined by its values over arbitrary max{width A,m} regular and under the first jet projection linearly independent elements of T _x ^A M. Further, we prove the rigidity of the so-called universally reparametrizable Weil algebras. Applying essentially those partial results we give the proof of the general rigidity result T ^A *M ? T ^r *M without coordinate computations, which improves and generalizes the partial result obtained in Tomá? (2009) from m ≥ k to all cases of m.We also introduce the space J ^A (M,N) of A-jets and prove its rigidity in the sense of its coincidence with the classical jet space J ^r (M,N).     

Mixed problem for the wave equation with integrable potential in the case of two-point boundary conditions of distinct orders

We study a mixed problem for the wave equation with integrable potential and with two-point boundary conditions of distinct orders for the case in which the corresponding spectral problem may have multiple spectrum. Based on the resolvent approach in the Fourier method and the Krylov convergence acceleration trick for Fourier series, we obtain a classical solution u(x, t) of this problem under minimal constraints on the initial condition u(x, 0) = ?(x). We use the Carleson–Hunt theorem to prove the convergence almost everywhere of the formal solution series in the limit case of ?(x) ∈ L _p[0, 1], p > 1, and show that the formal solution is a generalized solution of the problem.     

Inverse problem with nonlocal observation of finding the coefficient multiplying <Emphasis Type="Italic">u</Emphasis> <Subscript> <Emphasis Type="Italic">t</Emphasis> </Subscript> in the parabolic equation

We study the inverse problem of the reconstruction of the coefficient ?(x, t) = ?⁰(x, t) + r(x) multiplying u_t in a nonstationary parabolic equation. Here ?⁰(x, t) ≥ ?⁰ > 0 is a given function, and r(x) ≥ 0 is an unknown function of the class L_∞(Ω). In addition to the initial and boundary conditions (the data of the direct problem), we pose the problem of nonlocal observation in the form ∫₀^Tu(x, t) dμ(t) = χ(x) with a known measure dμ(t) and a function χ(x). We separately consider the case dμ(t) = ω(t)dt of integral observation with a smooth function ω(t). We obtain sufficient conditions for the existence and uniqueness of the solution of the inverse problem, which have the form of ready-to-verify inequalities. We suggest an iterative procedure for finding the solution and prove its convergence. Examples of particular inverse problems for which the assumptions of our theorems hold are presented.     

A unified approach to estimation of noncentrality parameters,the multiple correlation coefficient,and mixture models

We consider a class of mixture models for positive continuous data and the estimation of an underlying parameter θ of the mixing distribution. With a unified approach, we obtain classes of dominating estimators under squared error loss of an unbiased estimator, which include smooth estimators. Applications include estimating noncentrality parameters of chi-square and F-distributions, as well as ρ ²/(1 ? ρ ²), where ρ is amultivariate correlation coefficient in a multivariate normal set-up. Finally, the findings are extended to situations, where there exists a lower bound constraint on θ.