Relationship between Extremal and Sum Processes Generated by the same Point Process

2000 Mathematics Subject Classification: Primary 60G51, secondary 60G70, 60F17.We discuss weak limit theorems for a uniformly negligible triangular array (u.n.t.a.) in Z = [0, ∞) × [0, ∞)^d as well as for the associated with it sum and extremal processes on an open subset S . The complement of S turns out to be the explosion area of the limit Poisson point process. In order to prove our criterion for weak convergence of the sum processes we introduce and study sum processes over explosion area. Finally we generalize the model of u.n.t.a. to random sample size processes

Upper and Lower Bounds for Ruin Probability

In this note we discuss upper and lower bound for the ruin probability in an insurance model with very heavy-tailed claims and interarrival times

Effective optimization with weighted automata on decomposable trees

In this paper, we consider quantitative optimization problems on decomposable discrete systems. We restrict ourselves to labeled trees as the description of the systems and we use weighted automata on them as our computational model. We introduce a new kind of labeled decomposable trees, sum-like weighted labeled trees, and propose a method, which allows us to reduce the solution of an optimization problem, defined in a fragment of Weighted Monadic Second Order Logic, on such a tree to the solution of effectively derived problems, defined in the same logic, on its components with some additional post-processing. The approach originates from a method, proposed first by Feferman and Vaught in 1959, which we adapt and generalize. We adapt the method to the case of (fragments of) Weighted Monadic Second Order Logic and generalize it to the case of sum-like labeled trees rather than disjoint union and product. The main result of the paper may be applied in the wide range of optimization problems, such as critical path analysis or project planning, network optimization, generalized assignment problem, routing and scheduling as well as in the modern document languages like XML, image processing and compression, probabilistic systems or speech-to-text processing

Self-learning K-means clustering: a global optimization approach

An appropriate distance is an essential ingredient in various real-world learning tasks. Distance metric learning proposes to study a metric, which is capable of reflecting the data configuration much better in comparison with the commonly used methods. We offer an algorithm for simultaneous learning the Mahalanobis like distance and K-means clustering aiming to incorporate data rescaling and clustering so that the data separability grows iteratively in the rescaled space with its sequential clustering. At each step of the algorithm execution, a global optimization problem is resolved in order to minimize the cluster distortions resting upon the current cluster configuration. The obtained weight matrix can also be used as a cluster validation characteristic. Namely, closeness of such matrices learned during a sample process can indicate the clusters readiness; i.e. estimates the true number of clusters. Numerical experiments performed on synthetic and on real datasets verify the high reliability of the proposed method

Resampling approach for cluster model selection

In cluster analysis, selecting the number of clusters is an "ill-posed" problem of crucial importance. In this paper we propose a re-sampling method for assessing cluster stability. Our model suggests that samples' occurrences in clusters can be considered as realizations of the same random variable in the case of the "true" number of clusters. Thus, similarity between different cluster solutions is measured by means of compound and simple probability metrics. Compound criteria result in validation rules employing the stability content of clusters. Simple probability metrics, in particular those based on kernels, provide more flexible geometrical criteria. We analyze several applications of probability metrics combined with methods intended to simulate cluster occurrences. Numerical experiments are provided to demonstrate and compare the different metrics and simulation approaches

An estimate of the objective function optimum for the network Steiner problem

A complete weighted graph, (Formula presented.) , is considered. Let (Formula presented.) be some subset of vertices and, by definition, a Steiner tree is any tree in the graph G such that the set of the tree vertices includes set (Formula presented.). The Steiner tree problem consists of constructing the minimum-length Steiner tree in graph G, for a given subset of vertices (Formula presented.) The effectively computable estimate of the minimal Steiner tree is obtained in terms of the mean value and the variance over the set of all Steiner trees. It is shown that in the space of the lengths of the graph edges, there exist some regions where the obtained estimate is better than the minimal spanning tree-based one