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Author: Gennadiĭ Mikhaĭlovich Felʹdman
Publisher: American Mathematical Soc.
Published: 1993
Total Pages: 236
ISBN-13: 9780821845936
DOWNLOAD EBOOKThis book studies the problem of the decomposition of a given random variable introduction a sum of independent random variables (components). The central feature of the book is Feldman's use of powerful analytical techniques.
Author: Gennadij M. Fel'dman
Publisher: American Mathematical Soc.
Published:
Total Pages: 236
ISBN-13: 9780821897447
DOWNLOAD EBOOKThis book studies the problem of the decomposition of a given random variable introduction a sum of independent random variables (components). The central feature of the book is Feldman's use of powerful analytical techniques.
Author: Gennadĭ Mikaĭlovich Felʹdman
Publisher:
Published: 1993
Total Pages:
ISBN-13: 9781470445270
DOWNLOAD EBOOKThis book studies the problem of the decomposition of a given random variable into a sum of independent random variables (components). Starting from the famous Cramér theorem, which says that all components of a normal random variable are also normal random variables, the central feature of the book is Fel′′′dman's use of powerful analytical techniques. In the algebraic case, one cannot directly use analytic methods because of the absence of a natural analytic structure on the dual group, which is the domain of characteristic functions. Nevertheless, the methods developed in this book allow one.
Author: Gennadiy Feldman
Publisher: American Mathematical Society
Published: 2023-04-07
Total Pages: 253
ISBN-13: 1470472953
DOWNLOAD EBOOKIt 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.
Author: Gennadiĭ Mikhaĭlovich Felʹdman
Publisher: European Mathematical Society
Published: 2008
Total Pages: 272
ISBN-13: 9783037190456
DOWNLOAD EBOOKThis 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.
Author: Anatoli_ I_A_kovlevich Dorogovt_s_ev
Publisher: American Mathematical Soc.
Published: 2011-06-21
Total Pages: 362
ISBN-13: 0821868667
DOWNLOAD EBOOKThis book of problems is intended for students in pure and applied mathematics. There are problems in traditional areas of probability theory and problems in the theory of stochastic processes, which has wide applications in the theory of automatic control, queuing and reliability theories, and in many other modern science and engineering fields. Answers to most of the problems are given, and the book provides hints and solutions for more complicated problems.
Author: Mark Lʹvovich Agranovskiĭ
Publisher: American Mathematical Soc.
Published: 1993-01-01
Total Pages: 158
ISBN-13: 9780821897478
DOWNLOAD EBOOKThis book studies translation-invariant function spaces and algebras on homogeneous manifolds. The central topic is the relationship between the homogeneous structure of a manifold and the class of translation-invariant function spaces and algebras on the manifold. The author obtains classifications of translation-invariant spaces and algebras of functions on semisimple and nilpotent Lie groups, Riemann symmetric spaces, and bounded symmetric domains. When such classifications are possible, they lead in many cases to new characterizations of the classical function spaces, from the point of view of their group of admissible changes of variable. The algebra of holomorphic functions plays an essential role in these classifications when a homogeneous complex or $CR$-structure exists on the manifold. This leads to new characterizations of holomorphic functions and their boundary values for one- and multidimensional complex domains.
Author: Herbert Heyer
Publisher: World Scientific
Published: 2004
Total Pages: 399
ISBN-13: 9812562281
DOWNLOAD EBOOKThis book focuses on the algebraic-topological aspects of probabilitytheory, leading to a wider and deeper understanding of basic theorems, such as those on the structure of continuous convolution semigroupsand the corresponding processes with independent increments
Author: Masaya Yamaguchi
Publisher: American Mathematical Soc.
Published: 1997
Total Pages: 104
ISBN-13: 9780821805374
DOWNLOAD EBOOKThis book aims at providing a handy explanation of the notions behind the self-similar sets called "fractals" and "chaotic dynamical systems". The authors emphasize the beautiful relationship between fractal functions (such as Weierstrass's) and chaotic dynamical systems; these nowhere-differentiable functions are generating functions of chaotic dynamical systems. These functions are shown to be in a sense unique solutions of certain boundary problems. The last chapter of the book treats harmonic functions on fractal sets.
Author: O. V. Troshkin
Publisher: American Mathematical Soc.
Published: 1995-03-16
Total Pages: 212
ISBN-13: 9780821897614
DOWNLOAD EBOOKThis book discusses a number of qualitative features of mathematical models of incompressible fluids. Three basic systems of hydrodynamical equations are considered: the system of stationary Euler equations for flows of an ideal (nonviscous) fluid, stationary Navier-Stokes equations for flows of a viscous fluid, and Reynolds equations for the mean velocity field, pressure, and pair one-point velocity correlations of turbulent flows. The analysis concerns algebraic or geometric properties of vector fields generated by these equations, such as the general arrangement of streamlines, the character and distribution of singular points, conditions for their absence or appearance, and so on. Troshkin carries out a systematic application of the analysis to investigate conditions for unique solvability of a number of problems for these quasilinear systems. Containing many examples of particular phenomena illustrating the general ideas covered, this book will be of interest to researchers and graduate students working in mathematical physics and hydrodynamics.
