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Probability Theory: A Comprehensive Course (Universitext)

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Probability Theory: A Comprehensive Course (Universitext)

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    Available in PDF Format | Probability Theory: A Comprehensive Course (Universitext).pdf | English
    Achim Klenke(Author)

This second edition of the popular textbook contains a comprehensive course in modern probability theory. Overall, probabilistic concepts play an increasingly important role in mathematics, physics, biology,financial engineering and computer science. They help us in understanding magnetism, amorphous media, genetic diversity and the perils of random developments at financial markets, and they guide us in constructing moreefficient algorithms.
 
To address these concepts, the title covers a wide variety of topics, many of which are not usually found in introductory textbooks, such as:
 
• limit theorems forsums of random variables
• martingales
• percolation
• Markov chains and electrical networks
• construction of stochastic processes
• Poisson point process and infinitedivisibility
• large deviation principles and statistical physics
• Brownian motion
• stochastic integral and stochastic differential equations.

The theory is developedrigorously 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 probability theory. This second edition has beencarefully extended and includes many new features. It contains updated figures (over 50), computer simulations and some difficult proofs have been made more accessible. A wealth of examples and more than 270 exercises as wellas biographic details of key mathematicians support and enliven the presentation. It will be of use to students and researchers in mathematics and statistics in physics, computer science, economics and biology.

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*An electronic version of a printed book that can be read on a computer or handheld device designed specifically for this purpose.

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Book details

  • PDF | 652 pages
  • Achim Klenke(Author)
  • Springer; 2nd ed. 2014 edition (17 Sept. 2013)
  • English
  • 5
  • Science & Nature

Review Text

  • By An old man in the sea. on 22 July 2017

    Abstract: Lacks many theorems which could have made it a good reference, and too little intuition to be a textbook.Exercises are mere mathematical curiosities, at least in the first 4 chapters.I really doubt previous reviewers tried to learn from this book all by themselves. I've been trying to use this book for self-study, as I see no point to write a book which is cannot be used for that purpose, otherwise, I would just read the professors class notes or some other notes freely available on the internet.My main problem with this book is its lack of coherent flow. Let me give you an example. Let's consult the definition of distribution function. It's in page 26, and only for univariate domain. Until page 45, whenever a dist. function is mention is always in relation to a univariate domain. Well, in page 45, the author asks us to prove that a distribution function in R^2 must obey certain properties. Never does he define it explicitly... Yes, definition 1.103 allows implicitly for multivariate def., but the author never defines explicitly. It only makes the process of learning by this book much harder... This also points to another problem related to the exercises in this book (more on that later).Another example is that in page 39 he gives the Transformation formula for densities in R^n, but it's only in page 44 that he defines what is a density. Ok, this is one is not as problematic as the previous one, but both examples show just how the flow of the book is shattered.In the example of percolation, the author doesn't bother to give a description of the measure space, so when he proves that $\#C^p(x)$ is a random variable, it's a bit difficult to understand why. However, if you already know the details of percolation, why bother reading this section? You might just as well skip it.In chapter 5 he 'proves' Wald's Equality. In the 'proof', he write: «The same computation without absolute values yields the remaining part of the claim». What he doesn't tell you is that this computation is only valid because of Fubini Theorem, which is in chapter 14 !!!I could also give several other examples.The exercises have no sense of progression (from easiest to most difficult), and by solving them, one does not get the sense that he's increasing his knowledge of prob. theory in any way. The exercises seem more like mathematical curiosities, which are not that interesting, at least to someone who is learning the subject.Several simple theorems like «the limit of measurable functions is also measurable», are missing, and these theorems are very helpful in some exercises. For someone to learn probability theory from this book, it's simply too much wasted time researching the internet for missing details. We just get a spotted knowledge, albeit very 'comprehensive'.

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