Foundations of Modern Probability
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About the first edition:
To sum it up, one can perhaps see a distinction among advanced probability books into those which are original and path-breaking in content, such as Levy's and Doob's well-known examples, and those which aim primarily to assimilate known material, such as Loeve's and more recently Rogers and Williams'. Seen in this light, Kallenberg's present book would have to qualify as the assimilation of probability par excellence. It is a great edifice of material, clearly and ingeniously presented, without any non-mathematical distractions. Readers wishing to venture into it may do so with confidence that they are in very capable hands.
- Mathematical Reviews
This new edition contains four new chapters as well as numerous improvements throughout the text. There are new chapters on measure Theory-key results, ergodic properties of Markov processes and large deviations.
Reviews / Votes
"The book under review is the magnum opus of a brilliant scholar of probability. . An important feature of the book . is the writing style. . The choice of topics is excellent, some of which are not covered elsewhere. This book would make an outstanding text for graduate courses in measure theoretic probability, and for graduate students and faculty doing research in probability. The book is a delight to read. Highly recommended." (Myron Hlynka, Mathematical Reviews, April, 2022)
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Content
Introduction and Reading Guide.- I.Measure Theoretic Prerequisites: 1.Sets and functions, measures and integration.- 2.Measure extension and decomposition.- 3.Kernels, disintegration, and invariance.- II.Some Classical Probability Theory: 4.Processes, distributions, and independence.- 5.Random sequences, series, and averages.- 6.Gaussian and Poisson convergence.- 7.Infinite divisibility and general null-arrays.- III.Conditioning and Martingales: 8.Conditioning and disintegration.- 9.Optional times and martingales.- 10.Predictability and compensation.- IV.Markovian and Related Structures: 11.Markov properties and discrete-time chains.- 12.Random walks and renewal processes.- 13.Jump-type chains and branching processes.- V.Some Fundamental Processes: 14.Gaussian processes and Brownian motion.- 15.Poisson and related processes.- 16.Independent-increment and Lévy processes.- 17.Feller processes and semi-groups.- VI.Stochastic Calculus and Applications: 18.Itô integration and quadratic variation.- 19.Continuous martingales and Brownian motion.- 20.Semi-martingales and stochastic integration.- 21.Malliavin calculus.- VII.Convergence and Approximation: 22.Skorohod embedding and functional convergence.- 23.Convergence in distribution.- 24.Large deviations.- VIII.Stationarity, Symmetry and Invariance: 25.Stationary processes and ergodic theorems.- 26.Ergodic properties of Markov processes.- 27.Symmetric distributions and predictable maps.- 28.Multi-variate arrays and symmetries.- IX.Random Sets and Measures: 29.Local time, excursions, and additive functionals.- 30.Random mesures, smoothing and scattering.- 31.Palm and Gibbs kernels, local approximation.- X.SDEs, Diffusions, and Potential Theory: 32.Stochastic equations and martingale problems.- 33.One-dimensional SDEs and diffusions.- 34.PDE connections and potential theory.- 35.Stochastic differential geometry.- Appendices.- 1.Measurable maps.- 2.General topology.- 3.Linear spaces.- 4.Linear operators.- 5.Function and measure spaces.- 6.Classes and spaces of sets,- 7.Differential geometry.- Notes and References.- Bibliography.- Indices: Authors.- Topics.- Symbols.