Spectral theory and Wiener-Itô decomposition for the image of a Jacobi field

Assume that K ⁺: H _? → T _? is a bounded operator, where H _? and T _? are Hilbert spaces and ρ is a measure on the space H _?. Denote by ρ_K the image of the measure ρ under K ⁺. We study the measure ρ_K under the assumption that ρ is the spectral measure of a Jacobi field and obtain a family of operators whose spectral measure is equal to ρ_K. We also obtain an analog of the Wiener-Itô decomposition for ρ_K. Finally, we illustrate the results obtained by explicit calculations carried out for the case, where ρ_K is a Lévy noise measure.     

Non-Standard Skorokhod Convergence of Lévy-Driven Convolution Integrals in Hilbert Spaces

We study the convergence in probability in the non-standard M₁ Skorokhod topology of the Hilbert valued stochastic convolution integrals of the type to a process driven by a Lévy process L. In Banach spaces, we introduce strong, weak. and product modes of -convergence, prove a criterion for the -convergence in probability of stochastically continuous càdlàg processes in terms of the convergence in probability of the finite dimensional marginals and a good behavior of the corresponding oscillation functions, and establish criteria for the convergence in probability of Lévy driven stochastic convolutions. The theory is applied to the infinitely dimensional integrated Ornstein–Uhlenbeck processes with diagonalizable generators.     

On the Convergence of Chebyshev’s Method for Multiple Polynomial Zeros

In this paper we investigate the local convergence of Chebyshev’s iterative method for the computation of a multiple polynomial zero. We establish two convergence theorems for polynomials over an arbitrary normed field. A priori and a posteriori error estimates are also provided. All of the results are new even in the case of simple zero.     

Weak Convergence Results for Multiple Generations of a Branching Process

We establish limit theorems involving weak convergence of multiple generations of critical and supercritical branching processes. These results arise naturally when dealing with the joint asymptotic behavior of functionals defined in terms of several generations of such processes. Applications of our main result include a functional central limit theorem (CLT), a Darling–Erdös result, and an extremal process result. The limiting process for our functional CLT is an infinite dimensional Brownian motion with sample paths in the infinite product space (C ₀[0,1])^∞, with the product topology, or in Banach subspaces of (C ₀[0,1])^∞ determined by norms related to the distribution of the population size of the branching process. As an application of this CLT we obtain a central limit theorem for ratios of weighted sums of generations of a branching processes, and also to various maximums of these generations. The Darling–Erdös result and the application to extremal distributions also include infinite-dimensional limit laws. Some branching process examples where the CLT fails are also included.     

On Multiple Singular Integrals along Polynomial Curves with Rough Kernels

This paper is devoted to the study of a class of singular integral operators defined by polynomial mappings on product domains. Some rather weak size conditions, which imply the Lp boundedness of these singular integral operators as well as the corresponding maximal truncated singular integral operators for some fixed 1〈p〈 ∞,are given.     

Multiple G-Itô integral in G-expectation space

In 2007, Peng introduced Itô integral with respect to G-Brownian motion and the related Itô’s formula in G-expectation space. Motivated by the properties of multiple Wiener integral obtained by Itô in 1951, we introduce multiple G-Itô integral in G-expectation space, and investigate how to calculate it. Furthermore, We establish a relationship between Hermite polynomials and multiple G-Itô integrals.     

Lévy–LePage Series Representation of Stable Vectors: Convergence in Variation

Multidimensional stable laws G admit a well-known Lévy–LePage series representation

Fuzzy Integrals of Set-vatued Function

ＦｕｚｚｙＩｎｔｅｇｒａｌｓｏｆＳｅｔ－ｖａｌｕｅｄＦｕｎｃｔｉｏｎｓＺｈａｎｇＤｅｌｉａｎｄＷａｎｇＺｉｘｉａｏ（Ｄｅｐｔ．ｏｆＭａｔｈ．ＪｉｌｉｎＰｒｏｖ，Ｉｎｓｔ．ｏｆＥｄｕｃａｔｉｏｎＣｈａｎｇｃｈｕｎ，Ｊｉｌｉｎ，１３００２２，Ｐ．Ｒ．Ｃｈ...     

Integrals of psi—function

To the author's knowledge, among the so—called special functions, the gamma function is a unique one which is defined by a linear difference equation and is a hyper—transcendental function. There exists an another well—known hyper—transcendental function called the psi function, which is merely the logarithmic derivative of the gamma function. In this paper the author consider an extension of the gamma function and then obtain a series of integrals of the psi function.     

Pathwise Integrals and Itô–Tanaka Formula for Gaussian Processes

We prove an Itô–Tanaka formula and existence of pathwise stochastic integrals for a wide class of Gaussian processes. Motivated by financial applications, we define the stochastic integrals as forward-type pathwise integrals introduced by Föllmer and as pathwise generalized Lebesgue–Stieltjes integrals introduced by Zähle. As an application, we illustrate the importance of the Itô–Tanaka formula for pricing and hedging of financial derivatives.     

On Compositions of Generalized Fractional Integrals and Evaluation of Definite Integrals with Gauss Hypergeometric Functions

In this peper,composition formulas for generalized fractional integral oper-ators involving Gauss hypergeometric function are applied to evaluating of definite integrals involving two Gauss hypergeometric functions.     

On Compositions of Generalized Fractional Integrals

Compositions of generalized fractional integral operators involving Gauss hy-pergeometric function with power weights are studied.Com position formulas forsuch integrals which are the operators of the same type are obtained.In parti-cular,compositions of two identical operators are given.     

Differentiation of Integrals in RΩ

We show that the Lebesgue differentiation theorem does not hold in Ρ with the “product” Lebesgue measure. This revised version was published online in June 2006 with corrections to the Cover Date.     

Pointwise Convergence and Uniform Convergence of Wavelet Frame Series

Pointwise convergence and uniform convergence for wavelet frame series is a new topic. With the help of band-limited dual wavelet frames, this topic is first researched.