Our aim is to provide some convergence theorems for the Choquet integral with respect to various notions of convergence. For instance, the dominated convergence theorem for almost uniform convergence is related to autocontinuous set functions. Autocontinuity can also be related to convergence in measure, strict convergence or mean convergence. Whereas the monotone convergence theorem for almost...

Belief propagation is known to perform extremely well in many practical statistical inference and learning problems using graphical models, even in the presence of multiple loops. The use of the belief propagation algorithm on graphical models with loops is referred to as Loopy Belief Propagation (LBP). Various sufficient conditions for convergence of LBP have been presented; however, general n...

In the paper [9] we introduced the class of generalized metric spaces. These spaces simultaneously generalize ‘standard’ metric spaces, probabilistic metric spaces and fuzzy metric spaces. We show that every generalized metric space is, naturally, a uniform space. Thus we can use standard topological techniques to study, for example, probabilistic metric spaces. We illustrate this by proving a ...

This paper concerns the nonstandard problem of uniform deconvolution for nonperiodic functions over the real line. New algorithms are developed for this nonstandard statistical problem and integrated mean squared error bounds are established. We show that the upper bound of the integrated mean squared error for our new procedure is the same as for the standard case; hence these new estimators a...

In this paper we give a characterization of pointwise and uniform convergence of sequences of homogeneous polynomials on a Banach space by means of the convergence of their level sets. Results are obtained both in the real and the complex cases, as well as some generalizations to the nonhomogeneous case and to holomorphic functions in the complex case. Kuratowski convergence of closed sets is u...

Most studies were devoted to the design of efficient algorithms and the evaluation and application on diverse ranking problems, whereas few work has been paid to the theoretical studies on ranking learnability. In this paper, we study the relation between uniform convergence, stability and learnability of ranking. In contrast to supervised learning where the learnability is equivalent to unifor...

The problem of model selection in the face of nite sample size is considered within the framework of statistical decision theory. Focusing on the special case of regression, we introduce a model selection criterion which is shown to be robust in the sense that, with high conndence, even for a nite sample size it selects the best model. Our derivation is based on uniform convergence methods , au...

The goal of this work is the further development of neoclassical analysis, which extends the scope and results of the classical mathematical analysis by applying fuzzy logic to conventional mathematical objects, such as functions, sequences, and series. This allows us to reflect and model vagueness and uncertainty of our knowledge, which results from imprecision of measurement and inaccuracy of...