Laguerre Expansions of Gel′fand-Shilov Spaces

The subspaces G_α, G^β, and G^β_α (α, β ≥ 0)of Schwartz′ space S⁺ in (0, + ∞) are associated with the Hankel transform in the same way as the Gel′fand-Shilov spaces S_α, S^β, and S^β_α are associated with the Fourier transform. Indeed, if we consider the Hankel transform H_γ (γ < −1) defined by _γ(ƒ)(t) = ∫^∞₀ (xt)^−γ/2x^γJ_γ([formula]) ƒ(x) dx then _γ is an isomorphism from G_α, G^β, and G^β_α onto G^α, G_β, and G^α_β respectively. So. the spaces G^α_α are invariant for _γ. In this paper, we characterize the spaces G^α_α (α > 1) in terms of their Fourier-Laguerre coefficients. Also, we characterize the range of the Fourier-Laplace operator _D defined by _D(ƒ)(w) = ∫^∞₀ ƒ(t) e^{−(1/2)((1 + w)/(1 − w))t} for w D = {w : |w| ≤ 1} when it acts on the space G^α_α.     

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.     

Tractability of Multivariate Integration for Weighted Korobov Classes

We study the worst-case error of multivariate integration in weighted Korobov classes of periodic functions of d coordinates. This class is defined in terms of weights γ_j which moderate the behavior of functions with respect to successive coordinates. We study two classes of quadrature rules. They are quasi-Monte Carlo rules which use n function values and in which all quadrature weights are 1/n and rules for which all quadrature weights are non-negative. Tractability for these two classes of quadrature rules means that the minimal number of function values needed to guarantee error in the worst-case setting is bounded by a polynomial in d and ⁻¹. Strong tractability means that the bound does not depend on d and depends polynomially on ⁻¹. We prove that strong tractability holds iff ∑^∞_j=1 γ_j<∞, and tractability holds iff lim sup_d→∞ ∑^d_j=1 γ_j/log d<∞. Furthermore, strong tractability or tractability results are achieved by the relatively small class of lattice rules. We also prove that if ∑^∞_j=1 γ^1/α_j<∞, where α measures the decay of Fourier coefficients in the weighted Korobov class, then for d1, n prime and δ>0 there exist lattice rules that satisfy an error bound independent of d and of order n^−α/2+δ. This is almost the best possible result, since the order n^−α/2 cannot be improved upon even for d=1. A corresponding result is deduced for weighted non-periodic Sobolev spaces: if ∑^∞_j=1 γ^1/2_j<∞, then for d1, n prime and δ>0 there exist shifted lattice rules that satisfy an error bound independent of d and of order n^−1+δ. We also check how the randomized error of the (classical) Monte Carlo algorithm depends on d for weighted Korobov classes. It turns out that Monte Carlo is strongly tractable iff ∑^∞_j=1 log γ_j<∞ and tractable iff lim sup_d→∞ ∑^d_j=1 log γ_j/log d<∞. Hence, in particular, for γ_j=1 we have the usual Korobov space in which integration is intractable for the two classes of quadrature rules in the worst-case setting, whereas Monte Carlo is strongly tractable in the randomized setting.     

Summability of Product Jacobi Expansions

Orthogonal expansions in product Jacobi polynomials with respect to the weight function W_α, β(x)=∏^d_j=1 (1−x_j)^α_j (1+x_j)^β_j on [−1, 1]^d are studied. For α_j, β_j>−1 and α_j+β_j−1, the Cesàro (C, δ) means of the product Jacobi expansion converge in the norm of L^p(W_{α, β}, [−1, 1]^d), 1p<∞, and C([−1, 1]^d) if

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Moreover, for α_j, β_j−1/2, the (C, δ) means define a positive linear operator if and only if δ∑^d_i=1 (α_i+β_i)+3d−1.     

Stochastic comparisons of order statistics from heterogeneous populations, with applications in reliability

The basic result of the paper states: Let F₁, …, F_n, F₁^′,…, F_n^′ have proportional hazard functions with λ₁ ,…, λ_n , λ₁^′ ,…, λ_n^′ as the constants of proportionality. Let X₍₁₎ ≤ … ≤ X_(n) (X₍₁₎^′ ≤ … ≤ X_(n)^′) be the order statistics in a sample of size n from the heterogeneous populations {F₁ ,…, F_n}({F₁^′ ,…, F_n^′}). Then (λ₁ ,…, λ_n) majorizes (λ₁^′ ,…, λ_n^′) implies that (X₍₁₎ ,…, X_(n)) is stochastically larger than (X₍₁₎^′ ,…, X_(n)^′). Earlier results stochastically comparing individual order statistics are shown to be special cases. Applications of the main result are made in the study of the robustness of standard estimates of the failure rate of the exponential distribution, when observations actually come from a set of heterogeneous exponential distributions. Further applications are made to the comparisons of linear combinations of Weibull random variables and of binomial random variables.     

Maximum Likelihood Estimation of Means and Variances from Normal Populations Under Simultaneous Order Restrictions

For k normal populations with unknown means μ_i and unknown variances σ²_i, i = 1, ..., k, assume that there are some order restrictions among the means and variances, respectively, for example, simple order restrictions: μ₁ ≤ μ₂ ≤ ... ≤ μ_k and σ²₁ ≥ σ²₂ ≥ ... ≥ σ²_k > 0. Some properties of maximum likelihood estimation of μ_is and σ²_i are discussed and an algorithm of obtaining the maximum likelihood estimators under the order restrictions is proposed.     

Borel ideals vs. Borel sets of countable relations and trees

For every μ < ω₁, let I_μ be the ideal of all sets S ω^μ whose order type is <ω^μ. If μ = 1, then I₁ is simply the ideal of all finite subsets of ω, which is known to be Σ⁰₂-complete. We show that for every μ < ω₁, I_μ is Σ⁰_2μ-complete. As corollaries to this theorem, we prove that the set WO_ωμ of well orderings Rω × ω of order type <ω^μ is Σ⁰_2μ-complete, the set LP_μ of linear orderings R ω × ω that have a μ-limit point is Σ⁰_2μ+1-complete. Similarly, we determine the exact complexity of the set LT_μ of trees T ^<ωω of Luzin height <μ, the set WR_μ of well-founded partial orderings of height <μ, the set LR_μ of partial orderings of Luzin height <μ, the set WF_μ of well-founded trees T ^<ωω of height <μ(the latter is an old theorem of Luzin). The proofs use the notions of Wadge reducibility and Wadge games. We also present a short proof to a theorem of Luzin and Garland about the relation between the height of ‘the shortest tree’ representing a Borel set and the complexity of the set.     

Trace cocharacters and the Kronecker products of Schur functions

It follows from the theory of trace identities developed by Procesi and Razmyslov that the trace cocharacters arising from the trace identities of the algebra M_r(F) of r×r matrices over a field F of characteristic zero are given by TC_r,n=∑_λΛr(n)χ^λχ^λ where χ^λχ^λ denotes the Kronecker product of the irreducible characters of the symmetric group associated with the partition λ with itself and Λ_r(n) denotes the set of partitions of n with r or fewer parts, i.e. the set of partitions λ=(λ₁λ_k) with kr. We study the behavior of the sequence of trace cocharacters TC_r,n. In particular, we study the behavior of the coefficient of χ^(ν,n−m) in TC_r,n as a function of n where ν=(ν₁ν_k) is some fixed partition of m and n−mν_k. Our main result shows that such coefficients always grow as a polynomial in n of degree r−1.     

Nonnegative solutions of a nonlinear recurrence

Orthonormal polynomials with weight ¦τ¦ exp(−τ⁴) have leading coefficients with recurrence properties which motivate the more general equations ξ_m(ξ_{m − 1} + ξ_m + ξ_{m + 1}) = γ_m², M = 1, 2,…, where ξ_o is a fixed nonnegative value and γ₁, γ₂,… are positive constants. For this broader problem, the existence of a nonnegative solution is proved and criteria are found for its uniqueness. Then, for the motivating problem, an asymptotic expansion of its unique nonnegative solution is obtained and a fast computational algorithm, with error estimates, is given.     

On the Strong Law of Large Numbers and the Law of the Logarithm for Weighted Sums of Independent Random Variables with Multidimensional Indices

Let (X, X ; ^d} be a field of independent identically distributed real random variables, 0 < p < 2, and {a _, ; ( , ) ^d × ^d, ≤ } a triangular array of real numbers, where ^d is the d-dimensional lattice. Under the minimal condition that sup _, |a _, | < ∞, we show that | |^{− 1/p} ∑ _≤ a _, X → 0 a.s. as | | → ∞ if and only if E(|X|^p(L|X|)^{d − 1}) < ∞ provided d ≥ 2. In the above, if 1 ≤ p < 2, the random variables are needed to be centered at the mean. By establishing a certain law of the logarithm, we show that the Law of the Iterated Logarithm fails for the weighted sums ∑ _≤ a _, X under the conditions that EX = 0, EX² < ∞, and E(X²(L|X|)^{d − 1}/L₂|X|) < ∞ for almost all bounded families {a _, ; ( , ) ^d × ^d, ≤ of numbers.     

A one-parameter family of unimodal, concave, polynomial maps of the interval exhibiting multiplicities

This paper shows there exists a polynomial map, p, of the interval [0, 1] onto itself that is concave, symmetric about the point and such that, when parameterized {μp}, 0 ≤ μ ≤ 1, there exist three distinct values of the parameter μ₀ < μ₁ < μ₂ such that RLR³C = K(μ₀p) ≠ K(μ₁p) ≠ K(μ₂p) = RLR³C. There is also given an explicit construction of a C¹ family with the same properties.     

A generalization of the Weinstein-Moser theorems on periodic orbits of a Hamiltonian system near an equilibrium

We study the Hamiltonian system (HS) where H ε C² ( ^2N, ) satisfies H (0) = 0, H′ (0) = 0 and the quadratic form ) is non-degenerate. We fix τ₀ > 0 and assume that ²N E F decomposes into linear subspaces E and F which are invariant under the flow associated to the linearized system (LHS) = JH″ (0) x and such that each solution of (LHS) in E is τ₀-periodic whereas no solution of (LHS) in F − 0 is τ₀-periodic. We write σ(τ₀) = σ_Q(τ₀) for the signature of the quadratic form Q restricted to E. If σ(τ₀) ≠ 0 then there exist periodic solutions of (HS) arbitrarily close to 0. More precisely we show, either there exists a sequence x_k → 0 of τ_k-periodic orbits on the energy level H⁻¹ (0) with τ_k → τ₀; or for each λ close to 0 with λσ(τ₀) > 0 the energy level H⁻¹ (λ) contains at least distinct periodic orbits of (HS) near 0 with periods near τ₀. This generalizes a result of Weinstein and Moser who assumed QE to be positive definite.     

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.     

Asymptotic properties of Cramér-Smirnov statistics—A new approach

Let Y₁,…, Y_n be independent identically distributed random variables with distribution function F(x, θ), θ = (θ′₁, θ′₂), where θ_i (i = 1, 2) is a vector of p_i components, p = p₁ + p₂ and for θI, an open interval in ^p, F(x, θ) is continuous. In the present paper the author shows that the asymptotic distribution of modified Cramér-Smirnov statistic under H_n: θ₁ = θ₁₀ + n^−1/2γ, θ₂ unspecified, where γ is a given vector independent of n, is the distribution of a sum of weighted noncentral χ₁² variables whose weights are eigenvalues of a covariance function of a Gaussian process and noncentrality parameters are Fourier coefficients of the mean function of the Gaussian process. Further, the author exploits the special form of the covariance function by using perturbation theory to obtain the noncentrality parameters and the weights. The technique is applicable to other goodness-of-fit statistics such as U² [G. S. Watson, Biometrika 48 (1961), 109–114].     

Asymptotic Behavior of Heat Kernels on Spheres of Large Dimensions

Forn2, let (μ^x_τ, n)_τ0be the distributions of the Brownian motion on the unit sphereS^{n n+1}starting in some pointxSⁿ. This paper supplements results of Saloff-Coste concerning the rate of convergence ofμ^x_τ, nto the uniform distributionU_nonSⁿforτ→∞ depending on the dimensionn. We show that,[formula]forτ_n:=(ln n+2s)/(2n), where erf denotes the error function. Our proof depends on approximations of the measuresμ^x_τ, nby measures which are known explicitly via Poisson kernels onSⁿ, and which tend, after suitable projections and dilatations, to normal distributions on forn→∞. The above result as well as some further related limit results will be derived in this paper in the slightly more general context of Jacobi-type hypergroups.     

The Pointwise Densities of the Cantor Measure

Let be the classical middle-third Cantor set and let μ be the Cantor measure. Set s = log 2/log 3. We will determine by an explicit formula for every point x the upper and lower s-densities Θ*^s(μ , x), Θ*^s(μ , x) of the Cantor measure at the point x, in terms of the 3-adic expansion of x. We show that there exists a countable set F such that 9(Θ*^s(μ , x))^{− 1/s} + (Θ*^s(μ , x))^{− 1/s} = 16 holds for x \F. Furthermore, for μ_C almost all x, Θ*^s(μ , X) − 2 · 4^{− s} and Θ*^s(μ , x) = 4^{− s}. As an application, we will show that the s-dimensional packing measure of the middle-third Cantor set is 4^s.     

Estimation of a Mean Vector in a Two-Sample Problem

We consider the problem of estimating a p-dimensional vector μ₁ based on independent variables X₁, X₂, and U, where X₁ is N_p(μ₁, σ²Σ₁), X₂ is N_p(μ₂, σ²Σ₂), and U is σ²χ²_n (Σ₁ and Σ₂ are known). A family of minimax estimators is proposed. Some of these estimators can be obtained via Bayesian arguments as well. Comparisons between our results and the one of Ghosh and Sinha (1988, J. Multivariate Anal.27 206-207) are presented.     

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.     

Asymptotic distribution of the increase of the largest canonical correlation when one of the vectors is augmented

Let r₁ > r₂ > … be the sample canonical correlations in a sample of size n from a multivariate normal population partitioned into two subvectors with population canonical correlations ₁ > ₂ > …. Let one of the subvectors be augmented by adding one or more variables to it. For the increase in the largest canonical correlation, Δr in the sample and Δ in the population, it is shown that √n(Δr − Δ) → ^DN(0, σ²) and a formula for σ² is derived.     

Nonnegative Minimum Biased Quadratic Estimation in Mixed Linear Models

The problem of nonnegative quadratic estimation of a parametric function γ(β, σ)=β′Fβ+∑^r_i=1 f_iσ²_i in a general mixed linear model {y, Xβ, V(σ)=∑^r_i=1 σ²_iV_i} is discussed. Necessary and sufficient conditions are given for y′A₀y to be a minimum biased estimator for γ. It is shown how to formulate the problem of finding a nonnegative minimium biased estimator of γ as a conic optimization problem, which can be efficiently solved using convex optimization techniques. Models with two variance components are considered in detail. Some applications to one-way classification mixed models are given. For these models minimum biased estimators with minimum norms for square of expectation β² and for σ²₁ are presented in explicit forms.