Limit theorems for some polynomial statistics of the poisson process

The central limit theorem and the theorem on large deviations for the functionals of the Poisson random process are proved. The formulas for cumulants of multiple stochastic integrals (m.s.i.) with respect to the Poisson process are obtained. The m.s.i. may be considered as anU-statistics arising in queueing theory as well as a generalization of the well-known Poisson shot-noise process, having wide applications.     

Semicircular limits on the free Poisson chaos: Counterexamples to a transfer principle

We establish a class of sufficient conditions ensuring that a sequence of multiple integrals with respect to a free Poisson measure converges to a semicircular limit. We use this result to construct a set of explicit counterexamples showing that the transfer principle between classical and free Brownian motions (recently proved by Kemp, Nourdin, Peccati and Speicher (2012)) does not extend to the framework of Poisson measures. Our counterexamples implicitly use kernels appearing in the classical theory of random geometric graphs. Several new results of independent interest are obtained as necessary steps in our analysis, in particular: (i) a multiplication formula for free Poisson multiple integrals, (ii) diagram formulae and spectral bounds for these objects, and (iii) a counterexample to the general universality of the Gaussian Wiener chaos in a classical setting.     

Existence of Tangential Limits for α-Harmonic Functions on Half Spaces

Our aim in this paper is to prove the existence of tangential limits for Poisson integrals of the fractional order of functions in the L ^p Hölder space on half spaces.     

Regularity of the Laws of Shot Noise Series and of Related Processes

We investigate the regularity of shot noise series and of Poisson integrals. We give conditions for the absolute continuity of their law with respect to the Lebesgue measure and for their continuity in total-variation norm. In particular, the case of truncated series in addressed. Our method relies on a disintegration of the probability space based on a mere conditioning by the first jumps of the underlying Poisson process.     

Weighted Hardy spaces on an interval and Poisson integrals associated with ultraspherical series

We prove that certain maximal functions defined through the Poisson integrals associated with ultraspherical series characterize some weighted Hardy spaces on the finite interval.     

Riordan arrays associated with Laurent series and generalized Sheffer-type groups

A relationship between a pair of Laurent series and Riordan arrays is formulated. In addition, a type of generalized Sheffer groups is defined by using Riordan arrays with respect to power series with non-zero coefficients. The isomorphism between a generalized Sheffer group and the group of the Riordan arrays associated with Laurent series is established. Furthermore, Appell, associated, Bell, and hitting-time subgroups of the groups are defined and discussed. A relationship between the generalized Sheffer groups with respect to different type of power series is presented. The equivalence of the defined Riordan array pairs and generalized Stirling number pairs is given. A type of inverse relations of various series is constructed by using pairs of Riordan arrays. Finally, several applications involving various arrays, polynomial sequences, special formulas and identities are also presented as illustrative examples.     

The Finite Product Theorem for Certain Epireflections

We discuss a number of topics relating to multiple stochastic integration, where notions and ideas from point process theory seem particularly useful. Thus we give conditions for summability of certain multiple random series in terms of associated Poisson integrals, prove a decoupling result for divergence in probability to infinity, and give conditions for the existence of certain multiple integrals with respect to compensated POISSON and asymmetric LÉVY processes. In particular, the existence criteria for multiple p-stable integrals are shown to be independent of the skewness parameter.     

The approximate solution of a class of Fredholm integral equations with a weakly singular kernel

A method for finding the numerical solution of a weakly singular Fredholm integral equation of the second kind is presented. The Taylor series is used to remove singularity and Legendre polynomials are used as a basis. Furthermore, the Legendre function of the second kind is used to remove singularity in the Cauchy type integral equation. The integrals that appear in this method are computed in terms of gamma and beta functions and some of these integrals are computed in the Cauchy principal value sense without using numerical quadratures. Four examples are given to show the accuracy of the method.     

A note on the de La Vallée Poussin criterion for uniform integrability

In this paper, we present new versions of the classical de La Vallée Poussin criterion for uniform integrability. Our results concern the uniform integrability of a continuous function relative to a sequence of distribution functions. We apply our results to obtain a result on the convergence of a sequence of integrals which we illustrate with an example.     

Teh Bloch space of M-harmonic functions on bounded symmetric domains

New characterizations of Bloch functions on a bounded symmetric domain in ⁿ are given in terms of the intrinsic metrics. Various criteria for an M-harmonic function to be Bloch or little Bloch are obtained. These criteria establish the links between the M-harmonic Bloch and little Bloch functionsf and other notions including the Laplace-Betrami operator. We also study some basic properties of M-harmonic Bloch and little Bloch functions, extending previously known results for holomorphic functions. The space of M-harmonic functions contains the Poisson integrals of Borel measures and all pluriharmonic functions.     

Graded Poisson structures on the algebra of differential forms

We study the graded Poisson structures defined on Ω(M), the graded algebra of differential forms on a smooth manifoldM, such that the exterior derivative is a Poisson derivation. We show that they are the odd Poisson structures previously studied by Koszul, that arise from Poisson structures onM. Analogously, we characterize all the graded symplectic forms on ΩM) for which the exterior derivative is a Hamiltomian graded vector field. Finally, we determine the topological obstructions to the possibility of obtaining all odd symplectic forms with this property as the image by the pullback of an automorphism of Ω(M) of a graded symplectic form of degree 1 with respect to which the exterior derivative is a Hamiltonian graded vector field.     

Permutations avoiding consecutive patterns, II

The generating power series for the number of permutations avoiding particular consecutive patterns are derived in a new and simple fashion.Received: 24 March 2004; revised: 28 July 2004     

Energy-preserving methods for Poisson systems

We present and analyze energy-conserving methods for the numerical integration of IVPs of Poisson type that are able to preserve some Casimirs. Their derivation and analysis is done following the ideas of Hamiltonian BVMs (HBVMs) (see Brugnano et al. [10] and references therein). It is seen that the proposed approach allows us to obtain the methods recently derived in Cohen and Hairer (2011) [17], giving an alternative derivation of such methods and a new proof of their order. Sufficient conditions that ensure the existence of a unique solution of the implicit equations defining the formulae are given. A study of the implementation of the methods is provided. In particular, order and preservation properties when the involved integrals are approximated by means of a quadrature formula, are derived.     

Poisson Difference Schemes for Hamiltonian Systems on Poisson Manifolds

In this paper a systematical method for the construction of Poisson difference schemes with arbitrary order of accuracy for Hamiltonian systems on Poisson manifolds is considered. The transition of such difference schemes from one time-step to the next is a Poisson map. In addition, these schemes preserve all Casimir functions and, under certain conditions, quadratic first integrals of the original Hamiltonian systems. Especially, the arbitrary order centered schemes preserve all Casimir functions and all quadratic first integrals of the original Hamiltonian systems.     

Chaos decomposition of multiple fractional integrals and applications

Chaos decomposition of multiple integrals with respect to fractional Brownian motion (with H > 1/2) is given. Conversely the chaos components are expressed in terms of the multiple fractional integrals. Tensor product integrals are introduced and series expansions in those are considered. Strong laws for fractional Brownian motion are proved as an application of multiple fractional integrals. Received: 22 September 1998 / Revised version: 20 April 1999     

Poisson polyhedra in high dimensions

The zero cell of a parametric class of random hyperplane tessellations depending on a distance exponent and an intensity parameter is investigated, as the space dimension tends to infinity. The model includes the zero cell of stationary and isotropic Poisson hyperplane tessellations as well as the typical cell of a stationary Poisson Voronoi tessellation as special cases. It is shown that asymptotically in the space dimension, with overwhelming probability these cells satisfy the hyperplane conjecture, if the distance exponent and the intensity parameter are suitably chosen dimension-dependent functions. Also the high dimensional limits of the mean number of faces are explored and the asymptotic behaviour of an isoperimetric ratio is analysed. In the background are new identities linking the f-vector of the zero cell to certain dual intrinsic volumes.     

Multiple extensions of a finite Euler's pentagonal number theorem and the Lucas formulas

Motivated by the resemblance of a multivariate series identity and a finite analogue of Euler's pentagonal number theorem, we study multiple extensions of the latter formula. In a different direction we derive a common extension of this multivariate series identity and two formulas of Lucas. Finally we give a combinatorial proof of Lucas’ formulas.     

On a simple quasi-Monte Carlo approach for classical ultimate ruin probabilities

This note discusses a simple quasi-Monte Carlo method to evaluate numerically the ultimate ruin probability in the classical compound Poisson risk model. The key point is the Pollaczek–Khintchine representation of the non-ruin probability as a series of convolutions. Our suggestion is to truncate the series at some appropriate level and to evaluate the remaining convolution integrals by quasi-Monte Carlo techniques. For illustration, this approximation procedure is applied when claim sizes have an exponential or generalized Pareto distribution.     

Divergenzverhalten mehrdimensionaler Shannonscher Abtastreihen

The behaviour of multidimensional Shannon sampling series for continuous functions is examined. A continuous functiong ₁ εC ₀[0,1]² with support in the rectangle [0,1]×[0,1/2] is indicated in the paper for which the two dimensional Shannon sampling series diverge almost everywhere in the rectangle [0,1]×[1/2,1]. This shows that the localization principle for Shannon sampling series cannot hold in two dimensions and in higher dimensions. The result solves a problem formulated by P.L. Butzer.     

NOTE ON A BOUNDARY VALUE PROBLEM OF POISSON＇S EQUATION

This note shows that fur a BVP of Poisson‘s equation with Qm(u,v) as its source function, a direct evaluation of some integrals yields the same exact results as obtained by similarity analysis.