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Probability Theory: A Comprehensive Course de Achim Klenke

Probability Theory: A Comprehensive Course: Universitext

Autor Achim Klenke
en Limba Engleză Paperback – 16 ian 2008
Aimed primarily at graduate students and researchers, this text is a comprehensive course in modern probability theory and its measure-theoretical foundations. It covers a wide variety of topics, many of which are not usually found in introductory textbooks. The theory is developed rigorously and in a self-contained way, with the chapters on measure theory interlaced with the probabilistic chapters in order to display the power of the abstract concepts in the world of probability theory. In addition, plenty of figures, computer simulations, biographic details of key mathematicians, and a wealth of examples support and enliven the presentation.
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Specificații

ISBN-13: 9781848000476
ISBN-10: 1848000472
Pagini: 616
Ilustrații: 1
Dimensiuni: 159 x 232 x 28 mm
Greutate: 0.88 kg
Ediția:2008
Editura: Springer
Colecția Springer
Seria Universitext

Locul publicării:London, United Kingdom

Public țintă

Research

Descriere

Probabilistic concepts play an increasingly important role in mathematics, physics, biology, financial engineering and computer science. They help us to understand magnetism, amorphous media, genetic diversity and the perils of random developments on the financial markets, and they guide us in constructing more efficient algorithms.
This text is a comprehensive course in modern probability theory and its measure-theoretical foundations. Aimed primarily at graduate students and researchers, the book covers a wide variety of topics, many of which are not usually found in introductory textbooks, such as:
  • limit theorems for sums of random variables;
  • martingales;
  • percolation;
  • Markov chains and electrical networks;
  • construction of stochastic processes;
  • Poisson point processes and infinite divisibility;
  • large deviation principles and statistical physics;
  • Brownian motion; and
  • stochastic integral and stochastic differential equations.
The theory is developed rigorously and in a self-contained way, with the chapters on measure theory interlaced with the probabilistic chapters in order to display the power of the abstract concepts in the world of probability theory. In addition, plenty of figures, computer simulations, biographic details of key mathematicians, and a wealth of examples support and enliven the presentation.

Cuprins

Basic Measure Theory.- Independence.- Generating Functions.- The Integral.- Moments and Laws of Large Numbers.- Convergence Theorems.- Lp-Spaces and Radon-Nikodym Theorem.- Conditional Expectations.- Martingales.- Optional Sampling Theorems.- Martingale Convergence Theorems and their Applications.- Backwards Martingales and Exchangeability.- Convergence of Measures.- Probability Measures on Product Spaces.- Characteristics Functions and Central Limit Theorem.- Infinitely Divisible Distributions.- Markov Chains.- Convergence of Markov Chains.- Markov Chains and Electrical Networks.- Ergodic Theory.- Brownian Motion.- Law of the Iterated Logarithm.- Large Deviations.- The Poisson Point Process.- The Itô Integral.- Stochastic Differential Equations.- References.- Notation Index.- Name Index.- Subject Index.

Recenzii

From the reviews:
"The book is indeed comprehensive, consisting of 26 chapters on different topics. … can be well used as a reference book on a wide range of topics. The target audience is researchers and graduate students … . Numerous advanced topics are included, so that the book is more inclusive … . There is more than enough material for a two-semester course here. … the book will primarily be used as a reference book. For that purpose, it is a rich and relatively inexpensive choice." (Miklós Bóna, MathDL, January, 2008)
"This book of over 600 pages gives a self-contained presentation of modern probability theory. It is based on courses on advanced probability given by the author. … Most of the proofs are well detailed. … This book will be helpful for graduate students in mathematics … and for researchers in mathematics or theoretical physics." (Sophie Lemaire, Mathematical Reviews, Issue 2009 f)

Textul de pe ultima copertă

Probabilistic concepts play an increasingly important role in mathematics, physics, biology, financial engineering and computer science. They help us to understand magnetism, amorphous media, genetic diversity and the perils of random developments on the financial markets, and they guide us in constructing more efficient algorithms.

This text is a comprehensive course in modern probability theory and its measure-theoretical foundations. Aimed primarily at graduate students and researchers, the book covers a wide variety of topics, many of which are not usually found in introductory textbooks, such as:

  • limit theorems for sums of random variables;

  • martingales;

  • percolation;

  • Markov chains and electrical networks;

  • construction of stochastic processes;

  • Poisson point processes and infinite divisibility;

  • large deviation principles and statistical physics;

  • Brownian motion; and

  • stochastic integral and stochastic differential equations.

The theory is developed rigorously and in a self-contained way, with the chapters on measure theory interlaced with the probabilistic chapters in order to display the power of the abstract concepts in the world of probability theory. In addition, plenty of figures, computer simulations, biographic details of key mathematicians, and a wealth of examples support and enliven the presentation.

Caracteristici

Comprehensive and modern introduction to the most important fields of probability theory
Unique selection of topics, including many not usually found in introductory texts

Notă biografică

Achim Klenke is a professor at the Johannes Gutenberg University in Mainz, Germany. He is known for his work on interacting particle systems, stochastic analysis, and branching processes, in particular for his pioneering work with Leonid Mytnik on infinite rate mutually catalytic branching processes.