Products of Random Variables: Applications to Problems of Physics and to Arithmetical Functions

by Galambos; Janos

• ISBN13: 9780824754020
• ISBN10: 0824754026
• Format: Hardcover
• Copyright: 2004-07-20
• Publisher: CRC Press

Summary

Products of Random Variables explores the theory of products of random variables through from distributions and limit theorems, to characterizations, to applications in physics, order statistics, and number theory. It uses entirely probabilistic arguments in actualizing the potential of the asymptotic theory of products of independent random variables and obtaining results with dependent variables using a new Bonferroni-type argument. Systematically and comprehensively tracks the progression of research completed in the area over the last twenty years. Well-indexed and well-referenced, Products of Random Variables Clarifies foundational concepts such as symmetric and limiting distributions of products Examines various limit theorems, from logarithmically Poisson distributions to triangular arrays Explores characterization theorems, detailing normal, Cauchy, and bivariate distributions Describes models of interactive particles Elucidates dual systems of interactive particles, dual systems of increasing size, and random walks Covers the Kubilius-Turán inequality and distributions for multiplicative functions Probes sequences of prime divisors and prime numbers Discusses Markov chains, Hilbert spaces, and quotients of random variables Presents income growth models and numerous other applied models tapping products of random variables Authored by eminent scholars in the field, this volume is an important research reference for applied mathematicians, statisticians, physicists, and graduate students in these disciplines.

Author Biography

Janos Galambos is Professor of Mathematics at Temple University, Philadelphia, Pennsylvania Italo Simonelli is Associate Professor of Mathematics at Texas A&M University, Commerce, Texas

Table of Contents

Preface	iii
1 Foundations	1	(54)
1.1 Basic concepts and notation	1	(10)
1.1.1 Events and probability	1	(2)
1.1.2 Random variables	3	(3)
1.1.3 Weak convergence; characteristic functions	6	(2)
1.1.4 Convergence in probability and almost sure convergence	8	(3)
1.2 Elementary formulas for products	11	(4)
1.3 Mellin transform	15	(11)
1.4 Symmetric distributions	26	(13)
1.5 Bonferroni-type inequalities	39	(6)
1.6 Limiting distribution of products	45	(10)
2 Limit Theorems	55	(56)
2.1 Logarithmically Poisson distributions	55	(3)
2.2 The class of L-distributions for products	58	(4)
2.3 Absolute convergence of products	62	(8)
2.4 Conditional distributions for products	70	(9)
2.5 Triangular arrays	79	(21)
2.6 i.i.d. random variables	100	(3)
2.7 Order statistics as products	103	(8)
3 Characterization	111	(44)
3.1 General description of characterization	111	(3)
3.2 The Cauchy functional equation	114	(6)
3.3 Characterizations by decompositions	120	(15)
3.4 Normal and Cauchy distributions	135	(4)
3.5 The normal distribution	139	(8)
3.6 Bivariate distributions	147	(8)
4 Interacting particles	155	(44)
4.1 Introduction	155	(2)
4.2 Models and dual systems	157	(10)
4.3 IPS with dual of increasing size	167	(21)
4.4 Random Walks	188	(11)
5 Arithmetical functions	199	(50)
5.1 Introduction	199	(5)
5.2 The Kubilius-Turán inequality	204	(9)
5.3 A theorem of Bakštys	213	(17)
5.4 Distribution of multiplicative functions	230	(11)
5.5 Variants of a conjecture of Erdös	241	(2)
5.6 Sequences of prime divisors	243	(6)
6 Miscellaneous results	249	(54)
6.1 Binary variables	249	(4)
6.2 The sign of a product and Markov chains	253	(4)
6.3 Sequences of the prime numbers	257	(3)
6.4 Hilbert spaces	260	(7)
6.5 Applied models with products	267	(28)
6.5.1 Models related to income	267	(4)
6.5.2 Multiplying observations for smoothing	271	(2)
6.5.3 Estimation for the Pareto distribution	273	(1)
6.5.4 Products of beta random variables	274	(8)
6.5.5 Products of gamma random variables	282	(2)
6.5.6 Random discriminants	284	(2)
6.5.7 Normalized products of sums	286	(2)
6.5.8 Distribution of products of order statistics	288	(5)
6.5.9 Characterization of unimodality via products	293	(2)
6.6 Quotients of random variables	295	(8)
Bibliography	303	(14)
Author Index	317	(4)
Subject Index	321