Limit theorems for random processes with random time substitution

In this paper we prove a theorem on sufficient conditions for the convergence in the Skorokhod space D[0, 1] of a sequence of random processes with random time substitution. We obtain almost sure versions of this theorem.     

On r-quick limit sets for empirical and related processes based on mixing random variables

The r-quick limit points of normalized sample paths and empirical distribution functions of mixing processes are characterized. An r-quick version of Bahadur-Kiefer-type representation for sample quantiles is established, which yields the r-quick limit points of quantile processes. These results are applied to linear functions of order statistics. Some results on r-quick convergence of certain Gaussian processes are also established.     

On deviations between empirical and quantile processes for mixing random variables

Let {X_n} be a strictly stationary φ-mixing process with Σ_j=1^∞ φ^1/2(j) < ∞. It is shown in the paper that if X₁ is uniformly distributed on the unit interval, then, for any t [0, 1], |F_n⁻¹(t) − t + F_n(t) − t| = O(n^−3/4(log log n)^3/4) a.s. and sup_0≤t≤1 |F_n⁻¹(t) − t + F_n(t) − t| = (O(n^−3/4(log n)^1/2(log log n)^1/4) a.s., where F_n and F_n⁻¹(t) denote the sample distribution function and tth sample quantile, respectively. In case {X_n} is strong mixing with exponentially decaying mixing coefficients, it is shown that, for any t [0, 1], |F_n⁻¹(t) − t + F_n(t) − t| = O(n^−3/4(log n)^1/2(log log n)^3/4) a.s. and sup_0≤t≤1 |F_n⁻¹(t) − t + F_n(t) − t| = O(n^−3/4(log n)(log log n)^1/4) a.s. The results are further extended to general distributions, including some nonregular cases, when the underlying distribution function is not differentiable. The results for φ-mixing processes give the sharpest possible orders in view of the corresponding results of Kiefer for independent random variables.     

Hierarchies in φ‐spaces and applications

We establish some results on the Borel and difference hierarchies in φ‐spaces. Such spaces are the topological counterpart of the algebraic directed‐complete partial orderings. E.g., we prove analogs of the Hausdorff Theorem relating the difference and Borel hierarchies and of the Lavrentyev Theorem on the non‐collapse of the difference hierarchy. Some of our results generalize results of A. Tang for the space Pω. We also sketch some older applications of these hierarchies and present a new application to the question of characterizing the ω‐ary Boolean operations generating a given level of the Wadge hierarchy from the open sets. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

α-Stable limit theorems for sums of dependent random vectors

Several α-stable limit theorems for sums of dependent random vectors are proved via point processes theory; p-mixing, m-dependence, and the type of mixing treated within the extreme value theory are considered.     

On the ω-problem for some types of non-metrizable spaces

The ω-problem on a topological space X consists in finding out whether there exists a function whose oscillation is equal to a given upper semi-continuous (USC) function f:X→[0,∞] vanishing at isolated points of X. If such F exists, we call it an ω-primitive for f. Unlike the case of metrizable spaces, an ω-primitive need not exist if X is not metrizable. We study the ω-problem for f taking the value ∞ in the case of ordinal space, products of regular “constancy” spaces and the wedge sums of such spaces. Some open problems are formulated.     

Quadrature formulae connected to σ-orthogonal polynomials

Let dλ(t) be a given nonnegative measure on the real line , with compact or infinite support, for which all moments exist and are finite, and μ₀>0. Quadrature formulas of Chakalov–Popoviciu type with multiple nodes

where σ=σ_n=(s₁,s₂,…,s_n) is a given sequence of nonnegative integers, are considered. A such quadrature formula has maximum degree of exactness d_max=2∑_ν=1ⁿs_ν+2n−1 if and only if

The proof of the uniqueness of the extremal nodes τ₁,τ₂,…,τ_n was given first by Ghizzetti and Ossicini (Rend. Mat. 6(8) (1975) 1–15). Here, an alternative simple proof of the existence and the uniqueness of such quadrature formulas is presented. In a study of the error term R(f), an influence function is introduced, its relevant properties are investigated, and in certain classes of functions the error estimate is given. A numerically stable iterative procedure, with quadratic convergence, for determining the nodes τ_ν, ν=1,2,…,n, which are the zeros of the corresponding σ-orthogonal polynomial, is presented. Finally, in order to show a numerical efficiency of the proposed procedure, a few numerical examples are included.     

Criteria for the strong regularity of J-inner functions and γ-generating matrices

The class of left and right strongly regular J-inner mvf's plays an important role in bitangential interpolation problems and in bitangential direct and inverse problems for canonical systems of integral and differential equations. A new criterion for membership in this class is presented in terms of the matricial Muckenhoupt condition (A₂) that was introduced for other purposes by Treil and Volberg. Analogous results are also obtained for the class of γ-generating functions that intervene in the Nehari problem. The new criterion is simpler than the criterion that we presented earlier. A determinental criterion is also presented.     

Corrigendum to “Extending homeomorphisms from punctured surfaces to handlebodies” [Topology Appl. 155 (6) (2008) 610–621]

We correct the statements of Theorems 9 and 10 of [A. Cattabriga, M. Mulazzani, Extending homeomorphisms from punctured surfaces to handlebodies, Topology Appl. 155 (2008) 610–621], by adding missing generators, and improve the statement of Theorem 10, by removing some redundant generators.     

Convergence rates in the SLLN for some classes of dependent random fields

Let be a random field i.e. a family of random variables indexed by Nr, r?2. We discuss complete convergence and convergence rates under assumption on dependence structure of random fields in the case of nonidentical distributions. Results are obtained for negatively associated random fields, ρ^?-mixing random fields (having maximal coefficient of correlation strictly smaller then 1) and martingale random fields.     

Uniformly Gâteaux smooth approximations on

Every Lipschitz mapping from c₀(Γ) into a Banach space Y can be uniformly approximated by Lipschitz mappings that are simultaneously uniformly Gâteaux smooth and C^∞-Fréchet smooth.     

The Fréchet and limiting subdifferentials of integral functionals on the spaces

A new approach to computing the Fréchet subdifferential and the limiting subdifferential of integral functionals is proposed. Thanks to this way, we obtain formulae for computing the Fréchet and limiting subdifferentials of the integral functional , uL₁(Ω,E). Here is a measured space with an atomless σ-finite complete positive measure, E is a separable Banach space, and . Under some assumptions, it turns out that these subdifferentials coincide with the Fenchel subdifferential of F.     

On Bernstein–Markov-type inequalities for multivariate polynomials in -norm

Bernstein–Markov-type inequalities provide estimates for the norms of derivatives of algebraic and trigonometric polynomials. They play an important role in Approximation Theory since they are widely used for verifying inverse theorems of approximation. In the past decades these inequalities were extended to the multivariate setting, but the main emphasis so far was on the uniform norm. It is considerably harder to derive Bernstein–Markov-type inequalities in the L_q-norm, and it requires introduction of new methods. In this paper we verify certain Bernstein–Markov-type inequalities in L_q-norm on convex and star-like domains. Special attention is given to the question of how the geometry of the domain affects the corresponding estimates.     

Asymptotic structure and the existence of noncompact operators between Banach spaces

We investigate the problem of the existence of a noncompact operator T:X₀X→Y in terms of the asymptotic structure of separable Banach spaces X and Y. More precisely, for and , let T_ξ,η be the linear map which sends each x_i to y_i. We prove that if for some then every T:X₀X→Y is compact. If for n=2 all such maps have norm 1 we show the existence of a noncompact T:X₀X→Y.     

Gradient estimates and Jackson’s theorem in spaces related to measures

Jackson’s theorem is established in a new kind of holomorphic function space Q_μ related to measures in any starlike circular domain in . Particularly, the result covers many spaces including BMOA, Q_p, Q_K, and F(p,q,s) spaces in the unit ball of . Moreover, we construct integral operators which give pointwise estimates for the gradient of the difference in terms of the gradient on the boundary. The gradient estimates are independent of the measures in question and give rise to Jackson’s theorem.     

Successive iteration and positive symmetric solution for a Sturm–Liouville-like four-point boundary value problem with a -Laplacian operator

In this paper, we consider the following Sturm–Liouville-like four-point p-Laplacian boundary value problem with the nonlinear term f depending on the first-order derivative subject to the boundary conditions By using a monotone iterative technique, the existence of symmetric positive solutions and corresponding iterative schemes are obtained.     

An extension of the Wang transform derived from Bühlmann’s economic premium principle for insurance risk

It is well known that the Wang transform [Wang, S.S., 2002. A universal framework for pricing financial and insurance risks. Astin Bull. 32, 213–234] for the pricing of financial and insurance risks is derived from Bühlmann’s economic premium principle [Bühlmann, H., 1980. An economic premium principle. Astin Bull. 11, 52–60]. The transform is extended to the multivariate setting by [Kijima M., 2006. A multivariate extension of equilibrium pricing transforms: The multivariate Esscher and Wang transforms for pricing financial and insurance risks, Astin Bull. 36, 269–283]. This paper further extends the results to derive a class of probability transforms that are consistent with Bühlmann’s pricing formula. The class of transforms is extended to the multivariate setting by using a Gaussian copula, while the multiperiod extension is also possible within the equilibrium pricing framework.     

A regularization semismooth Newton method based on the generalized Fischer–Burmeister function for -NCPs

We consider a regularization method for nonlinear complementarity problems with F being a P₀-function which replaces the original problem with a sequence of the regularized complementarity problems. In this paper, this sequence of regularized complementarity problems are solved approximately by applying the generalized Newton method for an equivalent augmented system of equations, constructed by the generalized Fischer–Burmeister (FB) NCP-functions φ_p with p>1. We test the performance of the regularization semismooth Newton method based on the family of NCP-functions through solving all test problems from MCPLIB. Numerical experiments indicate that the method associated with a smaller p, for example p[1.1,2], usually has better numerical performance, and the generalized FB functions φ_p with p[1.1,2) can be used as the substitutions for the FB function φ₂.     

The Leray–Schauder approach to the degree theory for -perturbations of maximal monotone operators in separable reflexive Banach spaces

The purpose of this paper is to demonstrate the fact that the topological degree theory of Leray and Schauder may be used for the development of the topological degree theory for bounded demicontinuous (S₊)-perturbations f of strongly quasibounded maximal monotone operators T in separable reflexive Banach spaces. Certain basic homotopy properties and the extension of this degree theory to (possibly unbounded) strongly quasibounded perturbations f are shown to hold. This work uses the well known embedding of Browder and Ton, and extends the work of Berkovits who developed this theory for the case T=0. Besides being an interesting mathematical problem, the existence of such a degree theory may, conceivably, become useful in situations where the use of the Leray–Schauder degree (via infinite dimensional compactness) is necessary.