Onα-Symmetric Multivariate Characteristic Functions

Ann-dimensional random vector is said to have anα-symmetric distribution,α>0, if its characteristic function is of the form((|u₁|^α+…+|u_n|^α)^1/α). We study the classesΦ_n(α) of all admissible functions: [0, ∞)→ . It is known that members ofΦ_n(2) andΦ_n(1) are scale mixtures of certain primitivesΩ_nandω_n, respectively, and we show thatω_nis obtained fromΩ_2n−1byn−1 successive integrations. Consequently, curious relations between 1- and 2- (or spherically) symmetric distributions arise. An analogue of Askey's criterion gives a partial solution to a question of D. St. P. Richards: If(0)=1,is continuous, lim_t→∞ (t)=0, and⁽²ⁿ⁻²⁾(t) is convex, thenΦ_n(1). The paper closes with various criteria for the unimodality of anα-symmetric distribution.     

Summability of orthogonal expansions of several variables

Summability of spherical h-harmonic expansions with respect to the weight function ∏_j=1^d |x_j|^2κ_j (κ_j0) on the unit sphere S^d−1 is studied. The main result characterizes the critical index of summability of the Cesàro (C,δ) means of the h-harmonic expansion; it is proved that the (C,δ) means of any continuous function converge uniformly in the norm of C(S^d−1) if and only if δ>(d−2)/2+∑_j=1^d κ_j−min_1jd κ_j. Moreover, it is shown that for each point not on the great circles defined by the intersection of the coordinate planes and S^d−1, the (C,δ) means of the h-harmonic expansion of a continuous function f converges pointwisely to f if δ>(d−2)/2. Similar results are established for the orthogonal expansions with respect to the weight functions ∏_j=1^d |x_j|^2κ_j(1−|x|²)^μ−1/2 on the unit ball B^d and ∏_j=1^d x_j^κ_j−1/2(1−|x|₁)^μ−1/2 on the simplex T^d. As a related result, the Cesàro summability of the generalized Gegenbauer expansions associated to the weight function |t|^2μ(1−t²)^λ−1/2 on [−1,1] is studied, which is of interest in itself.     

Cesàro Asymptotics for Orthogonal Polynomials on the Unit Circle and Classes of Measures

The convergence in L²( ) of the even approximants of the Wall continued fractions is extended to the Cesàro–Nevai class CN, which is defined as the class of probability measures σ with lim_n→∞ ∑ⁿ⁻¹_k=0 |a_k|=0, {a_n}_n0 being the Geronimus parameters of σ. We show that CN contains universal measures, that is, probability measures for which the sequence {|_n|² dσ}_n0 is dense in the set of all probability measures equipped with the weak-* topology. We also consider the “opposite” Szeg class which consists of measures with ∑^∞_n=0 (1−|a_n|²)^1/2<∞ and describe it in terms of Hessenberg matrices.     

Precise asymptotics in some strong limit theorems for multidimensionally indexed random variables

Consider Z₊^d (d2)—the positive d-dimensional lattice points with partial ordering , let {X_k,kZ₊^d} be i.i.d. random variables with mean 0, and set S_n=∑_knX_k, nZ₊^d. We establish precise asymptotics for ∑_n|n|^r/p−2P(|S_n||n|^1/p), and for

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, (0δ1) as 0, and for

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as .     

On the Rate of Convergence of Two Bernstein–Bézier Type Operators for Bounded Variation Functions, II

The rates of convergence of two Bernstein–Bézier type operators B^(α)_n and L^(α)_n for functions of bounded variation have been studied for the case α1 by the author and A. Piriou (1998, J. Approx. Theory95, 369–387). In this paper the other case 0<α<1 is treated and asymptotically optimal estimations of B^(α)_n and L^(α)_n for functions of bounded variation are obtained. Besides, some interesting behaviors of the operators B^(α)_n and L^(α)_n (α>0) for monotone functions and functions of bounded variation are also given.     

Robust M-estimators of location vectors

Let X₁,…,X_n be i.i.d. random vectors in R^m, let θεR^m be an unknown location parameter, and assume that the restriction of the distribution of X₁−θ to a sphere of radius d belongs to a specified neighborhood of distributions spherically symmetric about 0. Under regularity conditions on and d, the parameter θ in this model is identifiable, and consistent M-estimators of θ (i.e., solutions of Σ_i=1ⁿψ(|X_i− |)(X_i− )=0) are obtained by using “re-descenders,” i.e., ψ's wh satisfy ψ(x)=0 for x≥c. An iterative method for solving for is shown to produce consistent and asymptotically normal estimates of θ under all distributions in . The following asymptotic robustness problem is considered: finding the ψ which is best among the re-descenders according to Huber's minimax variance criterion.     

Expansions for the multivariate chi-square distribution

Three classes of expansions for the distribution function of the χ_k²(d, R)-distribution are given, where k denotes the dimension, d the degree of freedom, and R the “accompanying correlation matrix.” The first class generalizes the orthogonal series with generalized Laguerre polynomials, originally given by Krishnamoorthy and Parthasarathy [12]. The second class contains always absolutely convergent representations of the distribution function by univariate chi-square distributions and the third class provides also the probabilities for any unbounded rectangular regions. In particular, simple formulas are given for the three-variate case including singular correlation matrices R, which simplify the computation of third order Bonferroni inequalities, e.g., for the tail probabilities of max{χ_i²|1 ≤ i ≤ k} (k > 3).     

On the Rate of Convergence of Two Bernstein–Bézier Type Operators for Bounded Variation Functions

In this paper we study the rate of convergence of two Bernstein–Bézier type operatorsB^(α)_nandL^(α)_nfor bounded variation functions. By means of construction of suitable functions and the method of Bojanic and Vuillemier (J. Approx. Theory31(1981), 67–79), using some results of probability theory, we obtain asymptotically optimal estimations ofB^(α)_nandL^(α)_nfor bounded variation functions at points of continuity and points of discontinuity.     

Some Asymptotic Formulae for Gaussian Distributions

This paper considers asymptotic expansions of certain expectations which appear in the theory of large deviation for Gaussian random vectors with values in a separable real Hilbert space. A typical application is to calculation of the “tails” of distributions of smooth functionals,p(r)=P{Φ(r⁻¹ξ)0},r→∞, e.g., the probability that a centered Gaussian random vector hits the exterior of a large sphere surrounding the origin. The method provides asymptotic formulae for the probability itself and not for its logarithm in a situation, where it is natural to expect thatp(r)=c′r^D exp{−c″r²}. Calculations are based on a combination of the method of characteristic functionals with the Laplace method used to find asymptotics of integrals containing a fast decaying function with “small” support.     

Normal Approximation Rate of the Kernel Smoothing Estimator in a Partial Linear Model

By establishing the asymptotic normality for the kernel smoothing estimatorβ_nof the parametric componentsβin the partial linear modelY=X′β+g(T)+, P. Speckman (1988,J. Roy. Statist. Soc. Ser. B50, 413–456) proved that the usual parametric raten^−1/2is attainable under the usual “optimal” bandwidth choice which permits the achievement of the optimal nonparametric rate for the estimation of the nonparametric componentg. In this paper we investigate the accuracy of the normal approximation forβ_nand find that, contrary to what we might expect, the optimal Berry–Esseen raten^−1/2is not attainable unlessgis undersmoothed, that is, the bandwidth is chosen with faster rate of tending to zero than the “optimal” bandwidth choice.     

α-Stable characterization of Banach spaces (1 < α < 2)

This work characterizes some subclasses of α-stable (0 < α < 1) Banach spaces in terms of the extendibility to Radon laws of certain α-stable cylinder measures. These result extend the work of S. Chobanian and V. Tarieladze (J. Multivar. Anal.7, 183–203 (1977)). For these spaces it is shown that every Radon stable measure is the continuous image of a stable measure on a suitable L_β space with β = α(1 − α)⁻¹. The latter result extends some work of Garling (Ann. Probab.4, 600–611 (1976)) and Jain (Proceedings, Symposia in Pure Math. XXXI, p. 55–65, Amer. Math. Soc., Providence, R.I.).     

A new approach to Euler splines

Starting from the exponential Euler polynomials discussed by Euler in “Institutions Calculi Differentialis,” Vol. II, 1755, the author introduced in “Linear operators and approximation,” Vol. 20, 1972, the so-called exponential Euler splines. Here we describe a new approach to these splines. Let t be a constant such that t=|t|e^iα, −π<α<π,t≠0,t≠1.. Let S₁(x:t) be the cardinal linear spline such that S₁(v:t) = t^v for all v ε Z. Starting from S₁(x:t) it is shown that we obtain all higher degree exponential Euler splines recursively by the averaging operation . Here S_n(x:t) is a cardinal spline of degree n if n is odd, while is a cardinal spline if n is even. It is shown that they have the properties S_n(v:t) = t^v for v ε Z.     

Minimax estimation of the mean of spherically symmetric distributions under general quadratic loss

For X one observation on a p-dimensional (p ≥ 4) spherically symmetric (s.s.) distribution about θ, minimax estimators whose risks dominate the risk of X (the best invariant procedure) are found with respect to general quadratic loss, L(δ, θ) = (δ − θ)′ D(δ − θ) where D is a known p × p positive definite matrix. For C a p × p known positive definite matrix, conditions are given under which estimators of the form δ_a,r,C,D(X) = (I − (ar(|X|²)) D^−1/2CD^1/2 |X|⁻²)X are minimax with smaller risk than X. For the problem of estimating the mean when n observations X₁, X₂, …, X_n are taken on a p-dimensional s.s. distribution about θ, any spherically symmetric translation invariant estimator, δ(X₁, X₂, …, X_n), with have a s.s. distribution about θ. Among the estimators which have these properties are best invariant estimators, sample means and maximum likelihood estimators. Moreover, under certain conditions, improved robust estimators can be found.     

Damped Oscillatory Integrals and the Boundedness Problem for Maximal Operators

Let S ⊂ ℝR ^{n +1} be a real-analytic hypersurface with surface measure dσ, and let ψ be a smooth nonnegative compactly supported cutoff function. Consider the surface measure dμ _q = ψ|Λ(X)|^q dσ, where Λ(X) is a damping factor determined by the matrices of the first and second fundamental forms of the surface. We show that its Fourier transform decays for large |ξ| as O (|ξ|^−(1/2+ε)), ε > 0, provided that q > 3/2. We also consider applications involving maximal operators associated with means of functions over hypersurfaces.__________Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 39, No. 2, pp. 70–74, 2005Original Russian Text Copyright © by I. A. Ikromov     

Ridge Functions and Orthonormal Ridgelets

Orthonormal ridgelets provide an orthonormal basis for L²(R²) built from special angularly-integrated ridge functions. In this paper we explore the relationship between orthonormal ridgelets and true ridge functions r(x₁ cos θ+x₂ sin θ). We derive a formula for the ridgelet coefficients of a ridge function in terms of the 1-D wavelet coefficients of the ridge profile r(t). The formula shows that the ridgelet coefficients of a ridge function are heavily concentrated in ridge parameter space near the underlying scale, direction, and location of the ridge function. It also shows that the rearranged weighted ridgelet coefficients of a ridge function decay at essentially the same rate as the rearranged weighted 1-D wavelet coefficients of the 1-D ridge profile r(t). In short, the full ridgelet expansion of a ridge function is in a certain sense equally as sparse as the 1-D wavelet expansion of the ridge profile. It follows that partial ridgelet expansions can give good approximations to objects which are countable superpositions of well-behaved ridge functions. We study the nonlinear approximation operator which “kills” coefficients below certain thresholds (depending on angular- and ridge-scale); we show that for approximating objects which are countable superpositions of ridge functions with 1-D ridge profiles in the Besov space B^1/p_p, p(R), 0<p<1, the thresholded ridgelet approximation achieves optimal rates of N-term approximation. This implies that appropriate thresholding in the ridgelet basis is equally as good, for certain purposes, as an ideally-adapted N-term nonlinear ridge approximation, based on perfect choice of N-directions.     

Asymptotic distributions of regression and autoregression coefficients with martingale difference disturbances

In this paper a form of the Lindeberg condition appropriate for martingale differences is used to obtain asymptotic normality of statistics for regression and autoregression. The regression model is y_t = Bz_t + v_t. The unobserved error sequence {v_t} is a sequence of martingale differences with conditional covariance matrices {Σ_t} and satisfying sup_{t=1,…, n} {v′_tv_tI(v′_tv_t>a) |z_t, v_t−1, z_t−1, …} 0 as a → ∞. The sample covariance of the independent variables z₁, …, z_n, is assumed to have a probability limit M, constant and nonsingular; max_t=1,…,nz′_tz_t/n 0. If (1/n)Σ_t=1ⁿΣ_t Σ, constant, then √nvec( _n−B) N(0,M⁻¹Σ) and _n Σ. The autoregression model is x_t = Bx_{t − 1} + v_t with the maximum absolute value of the characteristic roots of B less than one, the above conditions on {v_t}, and (1/n)Σ_t=max(r,s)+1(Σ_tv_t−1−rv′_t−1−s) δ_rs(ΣΣ), where δ_rs is the Kronecker delta. Then √nvec( _n−B) N(0,Γ⁻¹Σ), where Γ = Σ_{s = 0}^∞B^sΣ(B′)^s.     

Inequalities for a Polynomial and Its Derivative II

A well-known result of Rivlin states that if p(z) is a polynomial of degree n, such that p(z) ≠ 0 in |z| < 1, then max_{|z|=r < 1} |p(z)| ≤ ((r + 1)/2)ⁿ max_{|z| = 1} |p(z)|. In this paper, we consider the polynomial p(z) = a₀ + Σⁿ_{v = μ}a_υz^υ having all its zeros in |z| ≤ k > 1 and obtain a generalization of this result. Our result improves upon a result recently proved by Bidkham and Dewan (J. Math. Anal. Appl.166 (1992), 19-324).     

New Algorithm for Ordered Tree-to-Tree Correction Problem

The ordered tree-to-tree correction problem is to compute the minimum edit cost of transforming one ordered tree to another one. This paper presents a new algorithm for this problem. Given two ordered trees S and T, our algorithm runs in O(|S| |T| + min { ²_S|T| + ^2.5_{S T}, ²_T|S| + ^2.5_{T S}) time, where _S denotes the number of leaves of S and _S denotes the depth of S. The previous best algorithms for this problem run in O(|S| |T| min { _S, _S} min { _T, _T}) time (K. Zhang and D. Shasha, SIAM J. Comput.18, No. 6 (1989), 1245–1262) and in O(min {|S|²|T| log₂ |T|, |T|²|S| log₂ |S|}) time (P. N. Klein, in “Algorithms—ESA'98, 6th Annual European Symposium” (G. Bilardi, G. F. Italiano, A. Pietracaprina, and G. Pucci, Eds.), Lecture Notes in Computer Science, Vol. 1461, pp. 91–102, Springer-Verlag, Berlin/New York, 1998). As a comparison, our algorithm is asymptotically faster for certain kind of trees.     

Compact causal data interpolation

Consider a Hilbert space equipped with a time-structure, i.e., a resolution E of the identity on defined on subsets of some linearly ordered set Λ. For which x and y in is it possible to find a causal (time respecting) compact operator T, so that Tx = y? When T is required to be a Hilbert-Schmidt operator and (Λ, E) is sufficiently regular, this question is answered in terms of the “time-densities” of x and y. The condition is that the integral ∝_gLμ_x({s t})⁻¹ dμ_y(t) should be finite, where μ_x and μ_y are the measures on Λ given by μ_x(Ω) = ¦|E(Ω)x¦|² and μ_y(Ω) = ¦|E(Ω)y¦|². Further a solution is given for the related problem of minimizing the sum of ¦|Tx − y¦|² and the squared Hilbert-Schmidt norm ¦|R¦|₂² of T.     

Complex Roots of a Random Algebraic Polynomial

This paper, for any constantK, provides an exact formula for the average density of the distribution of the complex roots of equation η₀ + η₁z + η₂z² + ··· + η_{n − 1}z^{n − 1} = Kwhere η_j = a_j + ib_jand {a_j}^{n − 1}_{j = 0}and {b_j}^{n − 1}_{j = 0}are sequences of independent identically and normally distributed random variables andKis a complex number withKas its real and imaginary parts. The case of real roots of the above equation with real coefficients andK,z Ris well known. Further we obtain the limiting behaviour of this distribution function asntends to infinity.