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Softcover ISBN:  9781470472955 
Product Code:  SURV/273 
List Price:  $129.00 
MAA Member Price:  $116.10 
AMS Member Price:  $103.20 
eBook ISBN:  9781470473266 
Product Code:  SURV/273.E 
List Price:  $125.00 
MAA Member Price:  $112.50 
AMS Member Price:  $100.00 
Softcover ISBN:  9781470472955 
eBook ISBN:  9781470473266 
Product Code:  SURV/273.B 
List Price:  $254.00 $191.50 
MAA Member Price:  $228.60 $172.35 
AMS Member Price:  $203.20 $153.20 

Book DetailsMathematical Surveys and MonographsVolume: 273; 2023; 240 ppMSC: Primary 60; 62; 43; Secondary 39
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 KacBernstein theorem, is a typical example of a socalled 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 selfcontained 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.
ReadershipGraduate students and researchers interested in probability distributions or functional equations on groups.

Table of Contents

Chapters

Preliminaries

Independent random variables with independent sum and difference

Characterization of probability distributions through the independence of linear forms

Characterization of probability distributions through the symmetry of the conditional distribution of one linear form given another

Characterization theorems on the field of $p$adic numbers

Miscellaneous characterization theorems


Additional Material

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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 KacBernstein theorem, is a typical example of a socalled 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 selfcontained 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.
Graduate students and researchers interested in probability distributions or functional equations on groups.

Chapters

Preliminaries

Independent random variables with independent sum and difference

Characterization of probability distributions through the independence of linear forms

Characterization of probability distributions through the symmetry of the conditional distribution of one linear form given another

Characterization theorems on the field of $p$adic numbers

Miscellaneous characterization theorems