Weak Convergence of Distributions

This chapter introduces the fundamentals of weak convergence for real sequences. Definitions and examples are given. The Skorokhod representation theorem is proved and the chapter then considers the preservation of weak convergence under transformations. Next, the role of moments and characteristic functions is considered. In the leading case of random sums, the criteria for weak convergence and the concept of a stable distribution are studied.

Regression with Stable Errors Based on Order Statistics

Classical regression approaches are not robust when errors are heavy-tailed or asymmetric. That may be due to the non-existence of the mean or variance of the error distribution. Estimation based on trimmed data, which ignored outlier or leverage points, has an old history and frequently used. This procedure chooses fixed cut-off points. In this work, we use this idea recently applied for initial estimates of regression coefficients with heavy-tailed stable errors. We propose an effective procedure to calculate the cut-off points based on the tail index and skewness parameters of errors. We use the property of the existence of some moments of stable distribution order statistics. Data are trimmed based on ordered residuals of a least square regression. However, the trimmed data’s optimal number is determined based on the number of error order statistics whose variance exists. Then, we use the rest of the ordered data to estimate the regression coefficients. Based on these order statistics’ joint distribution, we analytically compute the bias and variance of the introduced estimator of regression parameters that was impossible for regression with stable errors.

A Total Lp-Norm Optimization for Bearing-Only Source Localization in Impulsive Noise with SαS Distribution

This paper considers the problem of robust bearing-only source localization in impulsive noise with symmetric α-stable distribution based on the Lp-norm minimization criterion. The existing Iteratively Reweighted Pseudolinear Least-Squares (IRPLS) method can be used to solve the least LP-norm optimization problem. However, the IRPLS algorithm cannot reduce the bias attributed to the correlation between system matrices and noise vectors. To reduce this kind of bias, a Total Lp-norm Optimization (TLPO) method is proposed by minimizing the errors in all elements of system matrix and data vector based on the minimum dispersion criterion. Subsequently, an equivalent form of TLPO is obtained, and two algorithms are developed to solve the TLPO problem by using Iterative Generalized Eigenvalue Decomposition (IGED) and Generalized Lagrange Multiplier (GLM), respectively. Numerical examples demonstrate the performance advantage of the IGED and GLM algorithms over the IRPLS algorithm.

Interaction of Non-derivative Vocabulary of Smolensk Dialect and the Belarusian Language: Full Lexical Correspondences

Specific non-derivative nouns that are functioning in Smolensk dialect reveal a significant number of lexical parallels in the independent languageе structure of the contemporary Belarusian language. The analysis of these words is carried out in order to establish their status in two synchronous lexical and gram- matical systems for further clarification relating to the number of lexical parallels of different origins in Smolensk dialect. The subject of the study is the volume of lexical meaning of specific nonderivative nouns recorded in independent, but at the same time contact-located language formations, in one of which they have a regional status (Smolensk dialect), and in the other one they are included in the codified form of the standard language (the Belarusian language). The relevance of the study is explained by the complex history of the Russian-Belarusian borderland, which is reflected both in Smolensk dialect and in the Belarusian language, which has been developed on the dialect basis. The common material and spiritual culture typical for the dialects of the Russian-Belarus-ian borderland is a marker revealing the specifics of the region described. Due to the common culture, these conditionally distant language units regularly coincide in the volume of lexical meaning and demonstrate a stable distribution within lexical and semantic groups in each language formation at the present stage of its functioning.