Decomposition of infinitely divisible characteristic functions with absolutely continuous poisson spectral measure

We shall consider the problem of characterizing infinitely divisible characteristic functions which have only infinitely divisible factors. Infinitely divisible characteristic functions treated in this paper are those which have absolutely continuous Poisson spectral measures and have no Gaussian component in their Lévy canonical representations. It will be shown that Ostrovskii's sufficient condition is also necessary in this case.     

Characterization of a class of infinitely divisible distributions in Hilbert space

It is established that the spectral measure of an infinitely divisible distribution F in a Hilbert space H is concentrated in a sphere of finite radius if and only if the integral ∫ _H exp (α∥x∥ In (∥x∥+1))dF is finite for some numberα>0. If this integral is finite for anyα>0 then the infinitely divisible distribution F is normal (maybe, degenerate).     

On the decomposition of infinitely divisible characteristic functions of several variables

Summary We prove here some theorems describing infinitely divisible characteristic functions defined on R ⁿ which have positive Poisson spectrum and belong to the class I ₀ of characteristic functions without indecomposable factor. These theorems are generalizations to the case of several variables of results due to I.V. Ostrovskiy in the case of one variable.This work was supported by the National Science Foundation under grant NSF-GP-6175.     

Decomposition of infinitely divisible characteristic functions without Gaussian component

Summary The problem of characterizing the infinitely divisible characteristic functions which have only infinitely divisible factors is considered. Under the assumption that both the absolutely continuous and the singular (or the discrete) components exist in Poisson spectral measures, several necessary conditions for this problem are obtained. These conditions admit partial converses and new examples of infinitely divisible characteristic functions which have only infinitely divisible factors are given.     

A generalization of the results of Pillai

In a recent article Pillai (1990,Ann. Inst. Statist. Math.,42, 157–161) showed that the distribution 1–E (–x ), 0<1; 0x, whereE (x) is the Mittag-Leffler function, is infinitely divisible and geometrically infinitely divisible. He also clarified the relation between this distribution and a stable distribution. In the present paper, we generalize his results by using Bernstein functions. In statistics, this generalization is important, because it gives a new characterization of geometrically infinitely divisible distributions with support in (0, ).     

Integral points on quadrics in three variables whose coordinates have few prime factors

The main theorem states that if f(x ₁, x ₂, x ₃) is an indefinite anisotropic integral quadratic form with determinant d(f), and t a non-zero integer such that d(f)t is square-free, then as long as there is one integer solution to f(x ₁, x ₂, x ₃) = t there are infinitely many such solutions for which the product x ₁ x ₂ x ₃ has at most 26 prime factors. The proof relies on the affine linear sieve and in particular automorphic spectral methods to obtain a sharp level of distribution in the associated counting problem. The 26 comes from applying the sharpest known bounds towards Selberg’s eigenvalue conjecture. Assuming the latter the number 26 may be reduced to 22.     

Complete rank theorem of advanced calculus and singularities of bounded linear operators

Let E and F be Banach spaces, f: U ⊂ E → F be a map of C ^r (r ⩾ 1), x ₀ ∈ U, and ft (x ₀) denote the FréLechet differential of f at x ₀. Suppose that f′(x ₀) is double split, Rank(f′(x ₀)) = ∞, dimN(f′(_{x 0})) > 0 and codimR(f′(x ₀)) s> 0. The rank theorem in advanced calculus asks to answer what properties of f ensure that f(x) is conjugate to f′(x ₀) near x ₀. We have proved that the conclusion of the theorem is equivalent to one kind of singularities for bounded linear operators, i.e., x ₀ is a locally fine point for f′(x) or generalized regular point of f(x); so, a complete rank theorem in advanced calculus is established, i.e., a sufficient and necessary condition such that the conclusion of the theorem to be held is given.      

Hilbert’s irreducibility theorem for prime degree and general polynomials

Letf (X, t)εℚ[X, t] be an irreducible polynomial. Hilbert’s irreducibility theorem asserts that there are infinitely manyt ₀εℤ such thatf (X, t ₀) is still irreducible. We say thatf (X, t) isgeneral if the Galois group off (X, t) over ℚ(t) is the symmetric group in its natural action. We show that if the degree off with respect toX is a prime ≠ 5 or iff is general of degree ≠ 5, thenf (X, t ₀) is irreducible for all but finitely manyt ₀εℤ unless the curve given byf (X, t)=0 has infinitely many points (x ₀,t ₀) withx ₀εℚ,t ₀εℤ. The proof makes use of Siegel’s theorem about integral points on algebraic curves, and classical results about finite groups, going back to Burnside, Schur, Wielandt, and others. Supported by the DFG.     

Extraction of smooth solutions of a class of regular equations in a rectangle

This work is devoted to the study of two-dimensional, regular, almost hypoelliptic operators P(D) = P(D ₂, D ₂) with regular Newton polyhedrons. It is proved that all generalized (weak) solutions of the equation P(D)u = f from a several weighted Sobolev space are infinitely differentiable functions in the rectangle {x ∈ E ²: −a < x ₁ < a, −b < x ₂ < b} in the variable x ₂, in which the function f is infinitely differentiable.     

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.     

Random matrix models of stochastic integral type for free infinitely divisible distributions

The Bercovici-Pata bijection maps the set of classical infinitely divisible distributions to the set of free infinitely divisible distributions. The purpose of this work is to study random matrix models for free infinitely divisible distributions under this bijection. First, we find a specific form of the polar decomposition for the Lévy measures of the random matrix models considered in Benaych-Georges [6] who introduced the models through their laws. Second, random matrix models for free infinitely divisible distributions are built consisting of infinitely divisible matrix stochastic integrals whenever their corresponding classical infinitely divisible distributions admit stochastic integral representations. These random matrix models are realizations of random matrices given by stochastic integrals with respect to matrix-valued Lévy processes. Examples of these random matrix models for several classes of free infinitely divisible distributions are given. In particular, it is shown that any free selfdecomposable infinitely divisible distribution has a random matrix model of Ornstein-Uhlenbeck type ?? ₀ ^?? e ^?1 d?? _t ^d , d ?? 1, where ?? _t ^d is a d × d matrix-valued Lévy process satisfying an I _log condition.     

Local conjugacy theorem, rank theorems in advanced calculus and a generalized principle for constructing Banach manifolds

Applications of locally fine property for operators are further developed. LetE andF be Banach spaces andF:U(x ₀)⊂E→F be C¹ nonlinear map, whereU (x ₀) is an open set containing pointx ₀∈E. With the locally fine property for Frechet derivativesf′(x) and generalized rank theorem forf′(x), a local conjugacy theorem, i. e. a characteristic condition forf being conjugate tof′(x ₀) near x₀,is proved. This theorem gives a complete answer to the local conjugacy problem. Consequently, several rank theorems in advanced calculus are established, including a theorem for C¹ Fredholm map which has been so far unknown. Also with this property the concept of regular value is extended, which gives rise to a generalized principle for constructing Banach submanifolds.     

On superpositions of continuous functions

We show that if Φ is an arbitrary countable set of continuous functions of n variables, then there exists a continuous, and even infinitely smooth, function ψ(x₁,...,x_n) such that ψ(x ₁, ...,x _n ) ?g [? (f ₁(x ₁, ... ,f _f (x _n ))] for any function ? from Φ and arbitrary continuous functions g and f_i, depending on a single variable.     

A nonstandard proof of a lemma from constructive measure theory

Suppose that f_n is a sequence of nonnegative functions with compact support on a locally compact metric space, that T is a nonnegative linear functional, and that T f_n < T f₀. A result of Bishop, foundational to a constructive theory of functional analysis, asserts the existence of a point x such that f_n (x) < f ₀(x). This paper extends this result to arbitrary Hausdorff spaces, and gives short proofs using nonstandard analysis. While such arguments used are not themselves constructive, they can illuminate where the difficulty lies in finding the point x. An algorithm for constructing x is then given, with a nonstandard proof that the algorithm converges to a correct value. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Multiple solutions of sublinear Lane-Emden elliptic equations

We study the sublinear elliptic equation, −Δ u = |u|^psgn u + f(x,u) in the bounded domain Ω under the zero Dirichlet boundary condition. We suppose that 0 < p < 1 and |f(x,u)| is small enough near u = 0 and do not suppose that f(x,u) is odd on u. Then we prove that this problem has infinitely many solutions. Supported in part by the Grant-in-Aid for Scientific Research (C) (No. 16540179), Ministry of Education, Culture, Sports, Science and Technology.     

Nonemptiness of Approximate Subdifferentials of Midpoint δ-Convex Functions

ABSTRACT

For some given positive δ, a function f:D ? X → ? is called midpoint δ-convex if it satisfies the Jensen inequality f[(x ₀ + x ₁)/2] ≤ [f(x ₀) + f(x ₁)]/2 for all x ₀, x ₁ ∈ D satisfying ‖x ₁ ? x ₀‖ ≥ δ (Hu, Klee, and Larman, SIAM J. Control Optimiz. Vol. 27, 1989). In this paper, we show that, under some assumptions, the approximate subdifferentials of midpoint δ-convex functions are nonempty.     

Infinitely divisible measures onp-adic groups

We show that any infinitely divisible measure on ap-adic algebraic group (p a prime) has a translatex, by an elementx centralizing the support of , such thatx can be embedded in a continuous real one-parameter semigroup {V _t } _t >0, asx=v ₁.     

Pre Stieltjes Functions

In this paper we characterize the class C_k{{\mathcal{C}_k}} of functions f on (0,∞) for which f(x), . . . ,(x ^k f(x))^(k) are completely monotonic for given k. In the limit we obtain the well-known characterization of the class of Stieltjes functions as those functions f defined on the positive half line for which (x ^k f(x))^(k) is completely monotonic on (0,∞) for all k ≥ 0.     

From Chaos to Global Convergence

The main purpose of this paper is to investigate dynamical systems F : \mathbbR² ? \mathbbR²{F : \mathbb{R}^2 \rightarrow \mathbb{R}^2} of the form F(x, y) = (f(x, y), x). We assume that f : \mathbbR² ? \mathbbR{f : \mathbb{R}^2 \rightarrow \mathbb{R}} is continuous and satisfies a condition that holds when f is non decreasing with respect to the second variable. We show that for every initial condition x₀ = (x ₀, y ₀), such that the orbit

Whitney theorem of interpolatory type for k-monotone functions

One of the main results of this paper is the following Whitney theorem of interpolatory type for k-monotone functions (i.e., functions f such that divided differences f[x ₀,…, x _k ] are nonnegative for all choices of (k + 1) distinct points x ₀,…, x _k .