Vague convergence of sums of independent random variables

A sequence (μ _n) of probability measures on the real line is said to converge vaguely to a measureμ if∫ fdμ _n →∫ fdμ for every continuous functionf withcompact support. In this paper one investigates problems analogous to the classical central limit problem under vague convergence. Let ‖μ‖ denote the total mass ofμ andδ ₀ denote the probability measure concentrated in the origin. For the theory of infinitesimal triangular arrays it is true in the present context, as it is in the classical one, that all obtainable limit laws are limits of sequences of infinitely divisible probability laws. However, unlike the classical situation, the class of infinitely divisible laws is not closed under vague convergence. It is shown that for every probability measureμ there is a closed interval [0,λ], [0,e ⁻¹] ⊂ [0,λ] ⊂ [0, 1], such thatβμ is attainable as a limit of infinitely divisible probability laws iffβ ε [0,λ]. In the independent identically distributed case, it is shown that if (x ₁ + ... +x _n)/a _n, a_n → ∞, converges vaguely toμ with 0<‖μ‖<1, thenμ=‖μ‖δ ₀. If furthermore the ratiosa _n+1/a _n are bounded above and below by positive numbers, thenL(x)=P[|X ₁|>x] is a slowly varying function ofx. Conversely, ifL(x) is slowly varying, then for everyβ ε (0, 1) one can choosea _n → ∞ so that the limit measure=βδ ₀. To the memory of Shlomo Horowitz This research was partially supported by the National Science Foundation.     

The Norm of the Laplace Operator Restriction to a Homogeneous Polynomial Space

We investigate the restriction Δ _r,μ of the Laplace operator Δ onto the space of r-variate homogeneous polynomials F of degree μ. In the uniform norm on the unit ball of ℝ ^r , and with the corresponding operator norm, ‖Δ _r,μ F‖≤‖Δ _r,μ ‖⋅‖F‖ holds, where, for arbitrary F, the ‘constant’ ‖Δ _r,μ ‖ is the best possible. We describe ‖Δ _r,μ ‖ with the help of the family T _μ (σ x), , of scaled Chebyshev polynomials of degree μ. On the interval [−1,+1], they alternate at least (μ−1)-times, as the Zolotarev polynomials do, but they differ from them by their symmetry. We call them Zolotarev polynomials of the second kind, and calculate ‖Δ _r,μ ‖ with their help. We derive upper and lower bounds, as well as the asymptotics for μ→∞. For r≥5 and sufficiently large μ, we just get ‖Δ _r,μ ‖=(r−2)μ(μ−1). However, for 2≤r≤4 or lower values of μ, the result is more complicated. This gives the problem a particular flavor. Some Bessel functions and the φcot φ-expansion are involved.      

On the second term in the spectral asymptotics of the maxwell operator in a filled resonator

It is shown that the characteristics of the Maxwell operator in a resonator with a smooth inhomogeneous anisotropic filler have constant multiplicity if and only if the matrices of dielectric permittivity and magnetic permeability are connected by the relation ε≡f μ, where f is a scalar-valued function. When ε≡f μ and the boundary is smooth and ideally conducting, the coefficient of λ² in the asymptotic expansion of the distribution function of the eigenvalues of the Maxwell operator turns out to be zero. When the multiplicity of the characteristics is variable, this coefficient can be either zero or nonzero. Bibliography: 22 titles. Translated fromProblemy Matematicheskogo Analiza, No. 13, 1992, pp. 48–79.     

Singular parts of pluriharmonic measures

A measure μ defined on the complex sphere S is called pluriharmonic if its Poisson integral is a pluriharmonic function (in the unit ball of ℂⁿ). A probability measure ρ is called representing if ∫_Sfdp=f(0) for all f in the ball algebra. It is shown that singular parts of pluriharmonic measures and representing measures are mutually singular. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 217, 1994, pp. 54–58.     

Orthogonal almost locally minimal projections on ℓ 1 n

A projectionP on a Banach spaceX with ‖P‖≤λ₀ is called almost locally minimal if, for every α>0 small enough, the ballB(P,α) in the space of operatorsL(X) does not contain a projectionQ with ‖Q‖≤‖P‖(1–Dα²), whereD=D(λ₀) is a constant independent of ‖P‖. It is shown that, for everyp≥1 and every compact abelian groupG, every translation invariant projection onL _p(G) is almost locally minimal. Orthogonal projections on ℓ ₁ ⁿ are investigated with respect to some weaker local minimality properties. Participant in Workshop in Linear Analysis and Probability, Texas A&M University, College Station, Texas 1998. Partially supported by the Edmund Landau Center for Research in Mathematical Analysis and Related Areas, sponsored by the Minerva Foundation (Germany).     

Implicit functions determined by equations with unbounded operators and the solvability of operator equations

For implicit functions determined by the equation T_x+F(λ,x)=0, where T is an unbounded operator, existence and differentiability conditions are established. As an application, the solvability conditions for equations of the form Tx+f(x)=0 are derived. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 248, 1998, pp. 216–224. Translated by L. Yu. Kolotilina.     

Coincidence criteria for the kernel of a function and the kernel of its integral almost positive means

We establish necessary and sufficient conditions for a point A of the Knopp kernelK(f) of a functionf to belong to the kernelK(M) of a functionM(t):=∫ _S fdμ _t , where the so-called almost positive measures μ _t determine a regular method of summation. In particular, this gives coincidence criteria for the kernelsK(f) andK(M). National Pedagogical University, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 9, pp. 1267–1275, September, 1999.     

On coerciveness and semiboundedness of general boundary problems

The paper treats coerciveness inequalities (of the form Re(Au, u)≧c ‖u‖ _s ² −λ ‖u‖ ₀ ² ,c>0,λ ∈ R) and semiboundedness inequalities (of the form Re (Au, u)≧−λ ‖u‖²) for the general boundary problems associated with an elliptic 2m-order differential operatorA in a compactn-dimensional manifold with boundary. In particular, we study the normal pseudo-differential boundary conditions, for which we determine necessary and sufficient conditions for coerciveness withs=m, and for semiboundedness with ‖u‖ = ‖u‖_m, in explicit form.     

An identity for normal-like operators

Forλεσ(A) (A a bounded linear operator on a Hilbert space) withλ a boundary point of the numerical range, the ‘spectral theory’ forλ is ‘just as ifA were normal’. IfA isnormal-like (the smallest disk containingσ(A) has radiusr=inf _z ‖A − z‖), then also sup {‖Ax‖² − |〈x.Ax〉|²:‖x‖=1}=r ². This research was partially supported by Air Force Contract AF-AFOSR-62-414.     

Onp-absolutely summing constants of banach spaces

Given 1≦p<∞ and a real Banach spaceX, we define thep-absolutely summing constantμ _p(X) as inf{Σ _i ^=1/m |x*(x _i)|^p p Σ _i ^=1/m ‖x _i‖^p p]¹ p}, where the supremum ranges over {x*∈X*; ‖x*‖≤1} and the infimum is taken over all sets {x ₁,x ₂, …,x _m} ⊂X such that Σ _i ^=1/m ‖x _i‖>0. It follows immediately from [2] thatμ _p(X)>0 if and only ifX is finite dimensional. In this paper we find the exact values ofμ _p(X) for various spaces, and obtain some asymptotic estimates ofμ _p(X) for general finite dimensional Banach spaces. This is a part of the author’s Ph.D. Thesis prepared at the Hebrew University of Jerusalem, under the supervision of Prof. A. Dvoretzky and Prof. J. Lindenstrauss.     

Spectral problems for pencils of polynomial matrices. Methods and algorithms. V

Methods and algorithms for the solution of spectral problems of singular and regular pencils D(λ, μ)=A(μ)-λB(μ) of polynomial matrices A(μ) and B(μ) are suggested (the separation of continuous and discrete spectra, the computation of points of a discrete spectrum with the corresponding, Jordan chains, the computation of minimal indices and a minimal basis of polynomial solutions, the computation of the determinant of a regular pencil). Bibliography: 13 titles. Translated by V. N. Kublanovskaya Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 202, 1992, pp. 26–70     

Some estimation theory on the sphere

Summary We consider distributions with densities of the formf(μ′x) andf(‖x _v ‖) where μ andx are unit vectors inR ^q and ‖x _v ‖ is the norm of the part ofx in somes dimensional subspaceV ofR ^q . For several loss functions, optimal Bayesian and Pitman estimators of μ andV are given. When uniform priors are used, these estimators are identical. Then the infinitesimal robustness characteristics of several special cases of these estimators are calculated.     

On oscillating kernels in the Besov space

In this paper we consider a convolution operator Tf=p.v. Ω * f with Ω(x)=K(x)×e^iλh(x), λ>0, where K(x) is a weak Calderón-Zygmund kernel and h(x) is a real-valued differentiable function. We give a boundedness criterion for such an operator to map the Besov space B ₁ ^0.1 (Rⁿ) into itself. This research was partially supported by NNSF and NEC in P. R. China.     

The bicompletion of an asymmetric normed linear space

A biBanach space is an asymmetric normed linear space (X,‖·‖) such that the normed linear space (X,‖·‖^s) is a Banach space, where ‖x‖^s= max {‖x‖,‖-x‖} for all x∈X. We prove that each asymmetric normed linear space (X,‖·‖) is isometrically isomorphic to a dense subspace of a biBanach space (Y,‖·‖_Y). Furthermore the space (Y,‖·‖_Y) is unique (up to isometric isomorphism). This revised version was published online in August 2006 with corrections to the Cover Date.     

A two-parameter boundary problem for second order differential system

Summary This paper is concerned with second order differential systems involving two parameters with boundary conditions specified at three points. In particular, we consider the system y' = k(x, λ, μ)z, z' = -g(x, λ, μ)y, where k and g are real-valued junctions defined on X: a ≤ x ≤ c, L: L₁ < λ < L₂, and M: M₁ < μ < M₂. This system is studied together with the boundary conditions α(λ, μ)y(a) - β(λ, μ)z(a)=0, γ(λ, μ)y(b) - δ(λ, μ)z(b)=0, ε₁(μ)y(b) - φ₁(μ)z(b)=ε₂(μ)y(c) - φ₂(μ)z(c), where α, β, δ, γ, ε_i, φ_i, i=1, 2, are continuous functions of the parameters. This work establishes the existence of eigenvalue pairs for the boundary problem and the oscillatory behavior of the associated solutions. These results complement those previously obtained by the authors and B. D. Sleeman, where boundary conditions of the ? Sturm-Liouville ? type were studied. Entrata in Redazione il 5 dicembre 1977. The research for this paper was supported by a University College Reasearch Grant, University of Alabama in Birmingham.     

On order preserving contractions

Let (Ω,Σ,μ) be a measure space and letP be an operator onL ₂(Ω,Σ,μ) with ‖P‖≦1,Pf≧0 a.e. wheneverf≧0. If the subspaceK is defined byK={x| ||P ⁿ x||=||P ^*n x||=||x||,n=1,2,...} thenK=L ₂(Ω,Σ₁,μ), where Σ₁ ⊂ Σ and onK the operatorP is “essentially” a measure preserving transformation. Thus the eigenvalues ofP of modulus one, form a group under multiplication. This last result was proved by Rota for finiteμ here finiteness is not assumed) and is a generalization of a theorem of Frobenius and Perron on positive matrices. The research reported in this document has been sponsored in part by Air Force Office of Scientific Research, OAR through the European Office, Aerospace Research, United States Air Force.     

Limit theorems for convolutions of probabilities on non abelian semigroups

Let S be a locally compact semigroup. We study the sequence (λⁿ) of the convolution powers of a probability measure λ on S and their shifts by a probability measure η on S. We shall give sufficient conditions for lim ‖λⁿ−η*λⁿ‖ = 0 (where ‖.‖ denotes the norm). In particular we consider the case the η is a point measure and we study the subsemigroup L_O(λ) = {x ∈ S : lim ‖λⁿ−δ_X*λⁿ‖ = 0}. We shall give necessary and sufficient conditions for L_o(λ)=S. In this case we want to treat the problem of the convergence of the sequence (λⁿ).     

Interior transition layers of solutions to the perturbed elliptic Sine-Gordon equation on an interval

We consider the perturbed elliptic Sine-Gordon equation on an interval-u″t+γsinu(t)=μf(u(t)),t ∈I := (-T, T),u(t) > 0,t ∈I,u(±T)=0 where λ, μ>0 are parameters andT>0 is a constant. By applying variational methods subject to the constraint depending on λ, we obtain eigenpairs (μ,u)=(μ(λ),u _λ) which solve this eigenvalue problem for a given λ>0. Then we study the asymptotic behavior ofu _λ and μ(λ) as λ→∞. Especially, we study the location of interior transition layers ofu _λ as λ→∞. This research has been supported by the Japan Society for the Promotion of Science.     

The spectrum of the averaging operator for a finite homogeneous graph

We give an estimate for the spectrum of the averaging operator T₁(Γ, 1) over the radius 1 for the finite (q+1)-homogeneous quotient graph Γ/X, where X is an infinite (q+1)-homogeneous tree associated with the free group G over a finite set of generators S={x₁ ..., x_p} (2p=q+1), and Γ, a subgroup of finite index in G. T₁(Γ, 1) is defined on the subspace L²(Γ/G, 1) ⊖ E_ex, where E_ex is the subspace of eigenfunctions of T₁(Γ, 1) with eigenvalue λ such that |λ|=q+1. We present a construction of some finite homogeneous graphs such that the spectrum of their adjacency matrices can be calculated explicitly. Bibliography: 11 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 205, 1993, pp. 92–109. Translated by A. M. Nikitin.     

A penalty method for nonparametric estimation of the logarithmic derivative of a density function

Summary Given a random sample of sizen from a densityf ₀ on the real line satisfying certain regularity conditions, we propose a nonparametric estimator forψ ₀=−f ₀ ^′ /f⁰. The estimate is the minimizer of a quadratic functional of the formλJ(ψ)+∫[ψ ²−2ψ′]dF_n where λ>0 is a smoothing parameter,J(·) is a roughness penalty, andF _n is the empirical c.d.f. of the sample. A characterization of the estimate (useful for computational purposes) is given which is related to spline functions. A more complete study of the caseJ(ψ)=∫[d ²ψ/dx²]² is given, since it has the desirable property of giving the maximum likelihood normal estimate in the infinite smoothness limit (λ→∞). Asymptotics under somewhat restrictive assumptions (periodicity) indicate that the estimator is asymptotically consistent and achieves the optimal rate of convergence. This type of estimator looks promising because the minimization problem is simple in comparison with the analogous penalized likelihood estimators. This research was supported by the Office of Naval Research under Grant Number N00014-82-C-0062.