Re: Loeve Probability Theory Pdf Download
Kirby Apodaca
Ding earned his Ph.D. in Statistics, focusing on probability theory in 2011 and currently serves as Chair Professor with the School of Mathematical Sciences at Peking University in Beijing. His research area is probability theory, focusing on interactions with statistical physics and computer science theory. In particular, his recent research topics include random constraint satisfaction problems, random planar geometry, Anderson localization, and disordered spin models.
The Liné and Michel Loève International Prize in Probability (Loève Prize) was created in 1992 in honor of Michel Loève by his widow, Liné. The prize, awarded every two years, is intended to recognize outstanding contributions by mathematical probability researchers under 45 years old, and comes with a $30,000 award.
My corollary. If $X_n$ are independent r.v.'s, then the set where $X_n$ converges and the set where $X_n$ diverges have probability 0 or 1; similarly for the series $\sum X_n$. Moreover, the if the limits of the sequences $X_n$ and $(X_1 + \ldots + X_n) / b_n$, where $b_n \uparrow \infty$, exist, then they are degenerate.
The mathematics of extreme events, or the remote parts of the probability distributions, is a discipline on its own, more important than any other with respect to risk and decisions since some domains are dominated by the extremes: for the class of subexponential (and of course for the subclass of power laws) the tails ARE the story.
The treatment has no measure theory, cuts to the chase, and can be used as a desk reference. If you want measure theory, go spend some time reading Billingsley. A deep understanding of measure theory is not necessary for scientific and engineering applications; it is not necessary for those who do not want to work on theorems and technical proofs.
I know which books I value when I end up buying a second copy after losing the first one. This book gives a complete overview of the basis of probability theory with some grounding in measure theory, and presents the main proofs. It is remarkable because of its concision and completeness: visibly prof Varadhan lectured from these notes and kept improving on them until we got this gem. There is not a single sentence too many, yet nothing is missing.
Varadhan has two other similar volumes one covering stochastic processes the other into the theory of large deviations, Large Deviations (Courant Lecture Notes) (though older than this current text). The book on Stochastic Processes, Stochastic Processes (Courant Lecture Notes) should be paired with this one.
"Every serious probabilist should, and doubtless will, possess a copy of this important work. Loève is to be complimented on completing his Herculean task at a uniformly high level of elegance." ? Journal of the American Statistical Association "This is a very scholarly book in the best tradition of analysis. Nothing else of this type exists for the benefit of the serious student of the subject and it is safe to predict that it will remain a standard compendium for many years to come." ? S. Vajda in Zentralblatt für Mathematik In the decades following its 1963 publication, this volume served as the standard advanced text in probability theory. Geared toward graduate students and professionals in the field of probability and statistics, the treatment offers extensive introductory material and is suitable for an undergraduate course in probability theory. The first four chapters cover notions of measure theory plus general concepts and tools of probability theory. Subsequent chapters explore sums of independent random variables, the central limit problem, conditioning, independence and dependence, ergodic theorems, and second order properties. The final two chapters examine foundations, martingales, and decomposability as well as Markov processes.
One of my advisers in graduate school was a probability theorist, as was hisadviser before him; I've not bothered to check, but I wouldn't be astonished ifthe chain went back to someone like Bernoulli. The fact that the chain couldgo back that far shows that mathematical probability is an old concept, almostas old as any other part of modern science; on the other hand, my adviser'sadviser came just after the generation, between the wars, which madeprobability a respectable and rigorous branch of mathematics and removedcountless obscurities from its applications. The first serious use ofstatistical methods in the sciences came only about a hundred years beforethat. Now of course error analysis is the first thing my students learn whenthey enter the lab. (Well, almost the first thing, after "if you don't writeit down, it never happened" and "Cosma can be bribed with chocolate.") I amconditioned to attack every problem as some kind of stochastic process; but afew generations back nobody had any but the vaguest idea what a stochasticprocess was.Pet peeves: Physicists who do not distinguish between a random variable ("X= the roll of a die") and the value it takes ("x=5"). People who reportestimated numbers without error-bars or confidence-intervals. Bayesians.
- See also:
- math in general
- stochastic processes
- statistics
- information theory
- algorithmic information theory
- statistical mechanics
- ergodic theory
- machine learning,statistical inference and induction
- dynamics
- large deviations
- empirical process theory
- concentration of measure
- deviation inequalities
- graph limits and exchangeable random graphs
- Hilbert Space Methods for Statistics and Probability
- projectivity in statistical models
- Recommended, big-picture:
- Patrick Billingsley, Probability and Measure
- Harald Cramér, Mathematical Methods ofStatistics [Review]
- William Feller, An Introduction to Probability Theory and ItsApplications, vol. I [I've not finished vol. II yet...]
- Bert Fristedt and Lawrence Gray, A Modern Approach toProbability Theory [Extremely thorough measure-theoretic text; nicetreatment of stochastic processes]
- Geoffrey Grimmett and David Stirzaker, Probability and RandomProcesses [Maybe the best contemporary textbook for those who do notneed measure-theoretic probability]
- Ian Hacking
- The Emergence of Probability [Where thatstrange two-faced notion came from, and why]
- The Taming of Chance [Putting chance to workin the 19th century]
- Recommended, close-ups, technical:
- Philippe Barbe, "An Elementary Approach to ExtremeValues Theory", arxiv:0811.0753
- Jochen Brocker, "A Lower Bound on Arbitrary f-Divergences inTerms of the Total Variation" arxiv:0903.1765
- H. E. Daniels, "Mixtures of Geometric Distributions",Journal of the Royal Statistical Society B 23 (1961): 409--413 [JSTOR]
- Alexander E. Holroyd and Terry Soo, "A Non-Measurable Set fromCoin-Flips", math.PR/0610705[A cute construction to help students see the point of measure-theoreticprobability]
- Mark Kac
- Selected Papers
- Statistical Independence in Probability, Analysis andNumber Theory
- Recommended, close-ups, conceptual and historical:
- Clark Glymour, "Instrumental Probability", Monist84 (2001): 284--300 [PDF reprint]
- Mark Kac, Engimas of Chance: An Autobiography
- Jill North, "Symmetry and Probability", phil-sci/2978
- Oystein Ore, Cardano, the Gambling Scholar
- Aris Spanos, "A frequentist interpretation of probability formodel-based inductiveinference", Synthese 190(2011) [With thanks to Prof. Spanos for letting me read a pre-publication draft]
- Jakob Rosenthal, "The Natural-Range Conception of Probability",phil-sci/4978[Defends the thesis that "the probability of an event is the proportion ofinitial states that lead to this event in the space of all possible initialstates, provided that this proportion is approximately the same in any not toosmall interval of the initial state space.... [I]n the types of situations thatgive rise to probabilistic phenomena we may expect to find an initial statespace such that any 'reasonable' density function over this space leads to thesame probabilities for the possible outcomes."]
- To read, historical:
- William J. Adams, The Life and Times of the CentralLimit Theorem
- Lorraine Daston, Classical Probability in theEnlightenment
- Hans Fischer, History of the Central Limit Theorem: From Laplace to Donsker
- Gerd Gigerenzer, Zeno Switjtink, Theodore Porter, Lorraine Daston,John Beatty and Lorenz Krüger, The Empire of Chance: How ProbabilityChanged Science and Everyday Life
- Kendall and Plackett (eds.), Studies in the History ofStatistics and Probability
- Andrei Kolmogorov, Foundations of Probability Theory
- Francesco Mainardi, Sergei Rogosin, "The origin of infinitely divisible distributions: from de Finetti's problem to Levy-Khintchine formula", arxiv:0801.1910
- Glenn Shafer and Vladimir Vovk, "The Sources of Kolmogorov'sGrundbegriffe", Statistical Science 21 (2006):70--98 = math.ST/0606533
- Reinhard Siegmund-Schultze
- "Probability in 1919/20: the von Mises-Pólya-Controversy", Archive for History of Exact Sciences 60 (2006): 431--515
- "A Non-Conformist Longing for Unity in the Fractures of Modernity: Towards a Scientific Biography of Richard von Mises (1883--1953)",Science in Context 17 (2004): 333--370
- To read, philosophical and foundational:
- Marshall Abrams, "Toward a Mechanistic Interpretation of Probability", phil-sci/4704
- Jordan Ellenberg and Elliott Sober,"Objective Probabilities in Number Theory" [PDF preprint]
- Eduardo M. R. A. Engel, A Road to Randomness in PhysicalSystems
- Alexander R. Pruss, "Probability, Regularity, and Cardinality",Philosophy of Science 80 (2013): 231--240
- John T. Roberts, "Laws About Frequencies", phil-sci/5058
- Michael Strevens, Bigger than Chaos: Understanding Complexitythrough Probability
- To read, pedagogical:
- Blom, Holst and Sandell, Problems and Snapshots from theWorld of Probability ["It is obvious that the authors have had fun inwriting this book..."]
- F. M. Dekking, C. Kraaikamp, H. P. Lopuhaä and L. E. Meester,A Modern Introduction to Probability and Statistics: Understanding Howand Why
- Feller, An Introduction to Probability Theory and ItsApplications vol. II
- Allan Gut, Probability: A Graduate Course [From theback: "'I know it's trivial, but I have forgotten why'. This is a slightlyexaggerated characterization of the unfortunate attitude of many mathematicianstoward the surrounding world. The point of departure of this book is theopposite. This textbook on the theory of probability is aimed at graduatestudents, with the ideology that rather than being a purely mathematicaldiscipline, probability theory is an intimate companion of statistics."]
- Svante Janson, "Probability asymptotics: notes on notation", arxiv:1108.3924 [Looks useful for the next time I teach stochastic processes]
- Emmanuel Lesigne, Heads or Tails: An Introduction to LimitTheorems in Probability
- Papoulis, Probability, Random Variables and StochasticProcesses
- Peter Olofsson, Probability, Statistics, and StochasticProcesses
- Sidney Resnick, A Probability Path
- A. Shiryaev, Probability Theory
- Stroock, Probability Theory: An Analytic View
- Paul Vitanyi, "Randomness," math.PR/0110086
- To read, technical:
- Sergio Albeverio and Song Liang, "Asymptotic expansions for theLaplace approximations of sums of Banach space-valued random variables", Annals ofProbability 33 (2005): 300--336 = math.PR/0503601
- David J. Aldous and Antar Bandyopadhyay, "A survey of max-typerecursive distributional equations", math.PR/0401388 = Annals of AppliedProbability 15 (2005): 1047--1110
- David Balding, Pablo A. Ferrari, Ricardo Fraiman and Mariela Sued,"Limit theorems for sequences of random trees", math.PR/0406280 [Abstract: "Weconsider a random tree and introduce a metric in the space of trees to definethe "mean tree" as the tree minimizing the average distance to the randomtree. When the resulting metric space is compact we show laws of large numbersand central limit theorems for sequence of independent identically distributedrandom trees. As application we propose tests to check if two samples of randomtrees have the same law." I wonder if the same technique could be applied toother kinds of random graphs, e.g., random scale-free networks?]
- Patrick Billinglsey, Convergence of Probability Measures
- Salomon Bochner, Harmonic Analysis and the Theory of Probability
- Tapas Kumar Chandra, The Borel-Cantelli Lemma
- Louis H. Y. Chen, Larry Goldstein and Qi-Man Shao, NormalApproximation by Stein's Method
- I. Calvo, J. C. Cuchí, J. G. Esteve, F. Falceto,"Generalized Central Limit Theorem and Renormalization Group", arxiv:1009.2899
- Sourav Chatterjee, "A new method of normal approximation",arxiv:math/0611213
- Bernard Chazelle, The Discrepency Method: Randomness andComplexity
- Irene Crimaldi and Luca Pratelli, "Two inequalities for conditionalexpectations and convergence results for filters", Statistics andProbability Letters 74 (2005): 151--162
- Victor De La Pena and Evarist Gine, Decoupling: FromDependence to Independence
- Victor H. de la Pena, Tze Leung Lai and Qi-Man Shan, Self-Normalized Processes: Limit Theory and Statistical Applications
- Janos Galambos and Italo Simonelli, Bonferroni-typeInequalities with Applications
- Stefano Galatolo, Mathieu Hoyrup, Cristobal Rojas, "A constructive Borel-Cantelli Lemma. Constructing orbits with required statistical properties", arxiv:0711.1478
- J. A. Gonzalez, L. I. Reyes, J. J. Suarez, L. E. Guerrero, andG. Gutierrez, "A mechanism for randomness," nlin.CD/0202022 [Color meskeptical, from the abstract]
- Martin Hairer, "A theory of regularity structures", arxiv:1303.5113
- Oliver Johnson and Andrew Barron, "Fisher Information inequalitiesand the Central Limit Theorem,"math.PR/0111020
- Oliver Johnson and Richard Samworth, "Central Limit Theorem andconvergence to stable laws in Mallows distance", math.PR/0406218
- Laurent Mazliak, "Poincarés Odds", arxiv:1211.5737
- Henry McKean, Probability: The Classical Limit Theorems
- National Research Council (USA), Probability andAlgorithms [online]
- Peter Orbanz, "Projective limit random probabilities on Polish spaces", Electronic Journal of Statistics5 (2011): 1354--1373
- Giovanni Peccati and Murad S. Taqqu, "Moments, cumulants and diagram formulae for non-linear functionals of random measures", arxiv:0811/1726
- Iosif Pinelis, "Between Chebyshev and Cantelli", arxiv:1011.6065
- Chris Preston, "Some notes on standard Borel and relatedspaces", arxiv:0809.3066
- Revesz, The Laws of Large Numbers
- R. Schweizer and A. Sklar, Probabilistic Metric Spaces
- Glenn Shafer and Vladimir Vovk, Probability and Finance: It'sOnly a Game! [Yet Another Foundation of Probability, this time fromgame-theory.]
- Akimichi Takemura, Vladimir Vovk, Glenn Shafer, "The generality of the zero-one laws", arxiv:0803.3679
- Ramon van Handel, Probability in High Dimension [PDF lecture notes]
- Roman Vershynin, High-Dimensional Probability: An Introduction with Applications in Data Science