Foundations of Modern Probability

Olav Kallenberg (Ph.D., Chalmers University, Gothenburg, Sweden, 1972) is an Emeritus Professor at Auburn University. He has held research positions in Sweden and abroad and taught in the US for more than 30 years. In 1977 he was awarded the Rollo Davidson Prize by Cambridge University, in 1989 he was elected a Fellow of the IMS, and in 1991–94 he served as the editor of PTRF. He has given plenary talks at major conferences all over the world, and was the opening lecturer at both the Vilnius Conference in 2006 and at the SPA Conference in Gothenburg in 2018. Apart from his numerous research papers, he is known for his Springer books Probabilistic Symmetries and Invariance Principles (2005) and Random Measures, Theory and Applications (2017). His book Foundations of Modern Probability (1997, 2002, and 2021) has become a classic reference work.

“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)

• 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.Stochasticdifferential 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.