Research Topics in Empirical Processes

<p>The first chapter consists of an overview of the theory of empirical processes, covering an introduction to empirical processes in R, uniform empirical processes and function parametric empirical processes in Section 1.1. Section 1.2 contains an overview of the theory related to the law of the iterated logarithm for Brownian motion and the modulus of continuity for Brownian motion. Section 1.3 contains the theory of the limiting processes for the empirical process, most importantly Brownian motion, Brownian bridge and the connections and relationships between them, with distributions of selected statistics of Brownian motion and Brownian bridge derived from reflection principles. Section 1.4 contains an overview of the theory required to prove central limit results for the empirical processes, covering the theory of the space C and Donsker’s theorem. The second chapter covers research topics, starting with Fourier analysis of mixture distributions and associated theory in Section 2.1. Section 2.2 covers findings in a research problem about non-linear autoregressive processes. Section 2.3 introduces a martingale approach to testing a regression model. Section 2.4 links the theory of ranks and sequential ranks to the theory of empirical processes.</p>

Research Topics in Empirical Processes

<p>The first chapter consists of an overview of the theory of empirical processes, covering an introduction to empirical processes in R, uniform empirical processes and function parametric empirical processes in Section 1.1. Section 1.2 contains an overview of the theory related to the law of the iterated logarithm for Brownian motion and the modulus of continuity for Brownian motion. Section 1.3 contains the theory of the limiting processes for the empirical process, most importantly Brownian motion, Brownian bridge and the connections and relationships between them, with distributions of selected statistics of Brownian motion and Brownian bridge derived from reflection principles. Section 1.4 contains an overview of the theory required to prove central limit results for the empirical processes, covering the theory of the space C and Donsker’s theorem. The second chapter covers research topics, starting with Fourier analysis of mixture distributions and associated theory in Section 2.1. Section 2.2 covers findings in a research problem about non-linear autoregressive processes. Section 2.3 introduces a martingale approach to testing a regression model. Section 2.4 links the theory of ranks and sequential ranks to the theory of empirical processes.</p>

Special empirical processes of independence indexed by classes of measurable functions

When analyzing statistical data in biomedical research, insurance, demography, as well as in other areas of practical research, random variables of interest take on certain values depending on the occurrence of certain events, e.g., when testing physical systems (individuals), the values of their operating time depend on the failures of subsystems; in the insurance business, the payments of insurance companies to their customers depends on insured events. In such experimental situations, the problems of studying the dependence of random variables on the corresponding events become rather important thus entailing the necessity to study the limiting properties of empirical processes indexed by classes of functions. The modern theory of empirical processes generalizes the classical results of the laws of large numbers, central and other limit theorems uniformly over the entire class of indexing under the imposition of entropy conditions. These theorems are generalized analogs of the classical theorems of Glivenko – Cantelli and Donsker. Special empirical processes are proposed in the study to check the independence of a random variable and an event. The properties of convergence of empirical processes to the corresponding Gaussian processes are analyzed. The results obtained are used to test the validity of the random right censoring model.

A descriptive phase model of problem-solving processes

Complementary to existing normative models, in this paper we suggest a descriptive phase model of problem solving. Real, not ideal, problem-solving processes contain errors, detours, and cycles, and they do not follow a predetermined sequence, as is presumed in normative models. To represent and emphasize the non-linearity of empirical processes, a descriptive model seemed essential. The juxtaposition of models from the literature and our empirical analyses enabled us to generate such a descriptive model of problem-solving processes. For the generation of our model, we reflected on the following questions: (1) Which elements of existing models for problem-solving processes can be used for a descriptive model? (2) Can the model be used to describe and discriminate different types of processes? Our descriptive model allows one not only to capture the idiosyncratic sequencing of real problem-solving processes, but simultaneously to compare different processes, by means of accumulation. In particular, our model allows discrimination between problem-solving and routine processes. Also, successful and unsuccessful problem-solving processes as well as processes in paper-and-pencil versus dynamic-geometry environments can be characterised and compared with our model.

A Dependent Lindeberg Central Limit Theorem for Cluster Functionals on Stationary Random Fields

In this paper, we provide a central limit theorem for the finite-dimensional marginal distributions of empirical processes (Zn(f))f∈F whose index set F is a family of cluster functionals valued on blocks of values of a stationary random field. The practicality and applicability of the result depend mainly on the usual Lindeberg condition and on a sequence Tn which summarizes the dependence between the blocks of the random field values. Finally, in application, we use the previous result in order to show the Gaussian asymptotic behavior of the proposed iso-extremogram estimator.

The Transformation and Enterprise Architecture Framework

This chapter proposes the applied holistic mathematical model for geopolitical analysis (AHMM4GA) that is the result of research on societal, business/financial, and geopolitical transformations using applied mathematical models. This research is based on an authentic and proprietary mixed research method that is supported by an underlining mainly qualitative holistic reasoning model module that punctually calls to quantitate functions. The proposed AHMM4GA formalism, attempts to simulate functions to support empirical processes.