LEADER 04102nam 2200565 450 001 996418201803316 005 20220321155619.0 010 $a3-030-56402-9 024 7 $a10.1007/978-3-030-56402-5 035 $a(CKB)5590000000005426 035 $a(MiAaPQ)EBC6382110 035 $a(DE-He213)978-3-030-56402-5 035 $a(MiAaPQ)EBC6647529 035 $a(Au-PeEL)EBL6382110 035 $a(OCoLC)1202751794 035 $a(Au-PeEL)EBL6647529 035 $a(PPN)255251467 035 $a(EXLCZ)995590000000005426 100 $a20220321d2020 uy 0 101 0 $aeng 135 $aurnn|008mamaa 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aProbability theory $ea comprehensive course /$fAchim Klenke 205 $aThird edition. 210 1$aCham, Switzerland :$cSpringer,$d[2020] 210 4$d©2020 215 $a1 online resource (XIV, 716 p. 55 illus., 24 illus. in color.) 225 1 $aUniversitext,$x0172-5939 311 $a3-030-56401-0 320 $aIncludes bibliographical references and index. 327 $a1 Basic Measure Theory -- 2 Independence -- 3 Generating Functions -- 4 The Integral -- 5 Moments and Laws of Large Numbers -- 6 Convergence Theorems -- 7 Lp-Spaces and the Radon?Nikodym Theorem -- 8 Conditional Expectations -- 9 Martingales -- 10 Optional Sampling Theorems -- 11 Martingale Convergence Theorems and Their Applications -- 12 Backwards Martingales and Exchangeability -- 13 Convergence of Measures -- 14 Probability Measures on Product Spaces -- 15 Characteristic Functions and the Central Limit Theorem -- 16 Infinitely Divisible Distributions -- 17 Markov Chains -- 18 Convergence of Markov Chains -- 19 Markov Chains and Electrical Networks -- 20 Ergodic Theory -- 21 Brownian Motion -- 22 Law of the Iterated Logarithm -- 23 Large Deviations -- 24 The Poisson Point Process -- 25 The Itô Integral -- 26 Stochastic Differential Equations -- References -- Notation Index -- Name Index -- Subject Index. 330 $aThis popular textbook, now in a revised and expanded third edition, presents a comprehensive course in modern probability theory. Probability plays an increasingly important role not only in mathematics, but also in physics, biology, finance and computer science, helping to understand phenomena such as magnetism, genetic diversity and market volatility, and also to construct efficient algorithms. Starting with the very basics, this textbook covers a wide variety of topics in probability, including many not usually found in introductory books, such as: limit theorems for sums of random variables martingales percolation Markov chains and electrical networks construction of stochastic processes Poisson point process and infinite divisibility large deviation principles and statistical physics Brownian motion stochastic integrals and stochastic differential equations. The presentation is self-contained and mathematically rigorous, with the material on probability theory interspersed with chapters on measure theory to better illustrate the power of abstract concepts. This third edition has been carefully extended and includes new features, such as concise summaries at the end of each section and additional questions to encourage self-reflection, as well as updates to the figures and computer simulations. With a wealth of examples and more than 290 exercises, as well as biographical details of key mathematicians, it will be of use to students and researchers in mathematics, statistics, physics, computer science, economics and biology. 410 0$aUniversitext,$x0172-5939 606 $aProbabilities 606 $aDistribution (Probability theory) 606 $aMeasure theory 615 0$aProbabilities. 615 0$aDistribution (Probability theory) 615 0$aMeasure theory. 676 $a519.2 700 $aKlenke$b Achim$0478404 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a996418201803316 996 $aProbability theory$9264282 997 $aUNISA