Functional Equations And Characterization Problems On Locally Compact Abelian Groups
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Functional Equations and Characterization Problems on Locally Compact Abelian Groups
Author: Gennadiĭ Mikhaĭlovich Felʹdman
language: en
Publisher: European Mathematical Society
Release Date: 2008
This book deals with the characterization of probability distributions. It is well known that both the sum and the difference of two Gaussian independent random variables with equal variance are independent as well. The converse statement was proved independently by M. Kac and S. N. Bernstein. This result is a famous example of a characterization theorem. In general, characterization problems in mathematical statistics are statements in which the description of possible distributions of random variables follows from properties of some functions in these variables. In recent years, a great deal of attention has been focused upon generalizing the classical characterization theorems to random variables with values in various algebraic structures such as locally compact Abelian groups, Lie groups, quantum groups, or symmetric spaces. The present book is aimed at the generalization of some well-known characterization theorems to the case of independent random variables taking values in a locally compact Abelian group $X$. The main attention is paid to the characterization of the Gaussian and the idempotent distribution (group analogs of the Kac-Bernstein, Skitovich-Darmois, and Heyde theorems). The solution of the corresponding problems is reduced to the solution of some functional equations in the class of continuous positive definite functions defined on the character group of $X$. Group analogs of the Cramer and Marcinkiewicz theorems are also studied. The author is an expert in algebraic probability theory. His comprehensive and self-contained monograph is addressed to mathematicians working in probability theory on algebraic structures, abstract harmonic analysis, and functional equations. The book concludes with comments and unsolved problems that provide further stimulation for future research in the theory.
Characterization of Probability Distributions on Locally Compact Abelian Groups
Author: Gennadiy Feldman
language: en
Publisher: American Mathematical Society
Release Date: 2023-04-07
It is well known that if two independent identically distributed random variables are Gaussian, then their sum and difference are also independent. It turns out that only Gaussian random variables have such property. This statement, known as the famous Kac-Bernstein theorem, is a typical example of a so-called characterization theorem. Characterization theorems in mathematical statistics are statements in which the description of possible distributions of random variables follows from properties of some functions of these random variables. The first results in this area are associated with famous 20th century mathematicians such as G. Pólya, M. Kac, S. N. Bernstein, and Yu. V. Linnik. By now, the corresponding theory on the real line has basically been constructed. The problem of extending the classical characterization theorems to various algebraic structures has been actively studied in recent decades. The purpose of this book is to provide a comprehensive and self-contained overview of the current state of the theory of characterization problems on locally compact Abelian groups. The book will be useful to everyone with some familiarity of abstract harmonic analysis who is interested in probability distributions and functional equations on groups.
What’s Past is Prologue
Author: Beth R. Bernhardt
language: en
Publisher: Purdue University Press
Release Date: 2018-11-15
Over one hundred presentations from the 37th annual Charleston Library Conference (held November 6–10, 2017) are included in this annual proceedings volume. Major themes of the meeting included data visualization, analysis and assessment of collections and library users, demand-driven acquisition, the future of print collections, and open access publishing. While the Charleston meeting remains a core one for acquisitions librarians in dialog with publishers and vendors, the breadth of coverage of this volume reflects the fact that this conference continues to be one of the major venues for leaders in the publishing and library communities to shape strategy and prepare for the future. Almost 2,000 delegates attended the 2017 meeting, ranging from the staff of small public library systems to the CEOs of major corporations. This fully indexed, copyedited volume provides a rich source for the latest evidence-based research and lessons from practice in a range of information science fields. The contributors are leaders in the library, publishing, and vendor communities.