Description
Springer Foundations Of Modern Probability 2 Vol Set 3rd Edition 2021 by Olav Kallenberg
The first edition of this single volume on the theory of probability has become a highly-praised standard reference for many areas of probability theory. Chapters from the first edition have been revised and corrected, and this edition contains four new chapters. New material covered includes multivariate and ratio ergodic theorems, shift coupling, Palm distributions, Harris recurrence, invariant measures, and strong and weak ergodicity. 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 Levy processes.- 17.Feller processes and semi-groups.- VI.Stochastic Calculus and Applications: 18.Ito 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.