Convergence for Sequences of Functions and an Egorov Type Theorem

For every ordinal <₁ we define a new type of convergence for sequences of functions (-uniform pointwise) which is intermediate between uniform and pointwise convergence. Using this type of convergence we obtain an Egorov type theorem for sequences of measurable functions.     

The notion of exhaustiveness and Ascoli-type theorems

In this paper we introduce the notion of exhaustiveness which applies for both families and nets of functions. This new notion is close to equicontinuity and describes the relation between pointwise convergence for functions and -convergence (continuous convergence). Using these results we obtain some Ascoli-type theorems dealing with exhaustiveness instead of equicontinuity. Also we deal with the corresponding notions of separate exhaustiveness and separate -convergence. Finally we give conditions under which the pointwise limit of a sequence of arbitrary functions is a continuous function.     

Basic matrix theorems for \mathcal{I}-convergence in (?)-groups

Some aspects of the theory of order and (D)-convergence in (?)-groups with respect to ideals are investigated. Moreover some new Basic Matrix Theorems are proved.     

Schur lemma and limit theorems in lattice groups with respect to filters

Some Schur, Nikodým, Brooks-Jewett and Vitali-Hahn-Saks-type theorems for (?)-group-valued measures are proved in the setting of filter convergence. Finally we pose an open problem.     

Ideal convergence and divergence of nets in (ℓ)-groups

In this paper we introduce the I- and I*-convergence and divergence of nets in (?)-groups. We prove some theorems relating different types of convergence/divergence for nets in (?)-group setting, in relation with ideals. We consider both order and (D)-convergence. By using basic properties of order sequences, some fundamental properties, Cauchy-type characterizations and comparison results are derived. We prove that I*-convergence/divergence implies I-convergence/divergence for every ideal, admissible for the set of indexes with respect to which the net involved is directed, and we investigate a class of ideals for which the converse implication holds. Finally we pose some open problems.     

Some versions of limit and Dieudonné-type theorems with respect to filter convergence for (<Emphasis Type="Italic">ℓ</Emphasis>)-group-valued measures

Some limit and Dieudonné-type theorems in the setting of (ℓ)-groups with respect to filter convergence are proved, extending earlier results.     

A Topological View of Ramsey Families of Finite Subsets of Positive Integers

If is an initially hereditary family of finite subsets of positive integers (i.e., if and G is initial segment of F then ) and M an infinite subset of positive integers then we define an ordinal index . We prove that if is a family of finite subsets of positive integers such that for every the characteristic function χ_F is isolated point of the subspace

Estimating the Distribution Function of a Stationary Process Involving Measurement Errors

A nonparametric estimation of a distribution function is considered when observations contain measurement errors. A method is developed to establish asymptotic normality results for a deconvoluting kernel-type estimator for ρ-mixing stochastic processes corrupted by some noise process. It is shown that the asymptotic distribution depends on the smoothness of the noise distributions, which are characterized as either ordinary smooth or super smooth. Also, the kind of dependence of the noise process is crucial to the form of the asymptotic variance. This revised version was published online in June 2006 with corrections to the Cover Date.     

p-convergence in measure of a sequence of measurable functions and corresponding minimal elements of c 0

We characterize convergence in measure of a sequence (f_n)_n of measurable functions to a measurable function f by elements of c₀, which express the quality of convergence of (f_n)_n to f. This characterization motivates the introduction of a new notion of convergence, called “p-convergence in measure” (p > 0), which is stronger than convergence in measure. We prove the existence of “minimal” elements in c₀ which characterize the convergence in measure of (f_n)_n to f.