*Stéphane Boucheron, Gábor Lugosi, and Pascal Massart*

- Published in print:
- 2013
- Published Online:
- May 2013
- ISBN:
- 9780199535255
- eISBN:
- 9780191747106
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199535255.001.0001
- Subject:
- Mathematics, Probability / Statistics, Applied Mathematics

This monograph presents a mathematical theory of concentration inequalities for functions of independent random variables. The basic phenomenon under investigation is that if a function of many ...
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This monograph presents a mathematical theory of concentration inequalities for functions of independent random variables. The basic phenomenon under investigation is that if a function of many independent random variables does not depend too much on any of them then it is concentrated around its expected value. This book offers a host of inequalities to quantify this statement. The authors describe the interplay between the probabilistic structure (independence) and a variety of tools ranging from functional inequalities, transportation arguments, to information theory. Applications to the study of empirical processes, random projections, random matrix theory, and threshold phenomena are presented. The book offers a self-contained introduction to concentration inequalities, including a survey of concentration of sums of independent random variables, variance bounds, the entropy method, and the transportation method. Deep connections with isoperimetric problems are revealed. Special attention is paid to applications to the supremum of empirical processes.Less

This monograph presents a mathematical theory of concentration inequalities for functions of independent random variables. The basic phenomenon under investigation is that if a function of many independent random variables does not depend too much on any of them then it is concentrated around its expected value. This book offers a host of inequalities to quantify this statement. The authors describe the interplay between the probabilistic structure (independence) and a variety of tools ranging from functional inequalities, transportation arguments, to information theory. Applications to the study of empirical processes, random projections, random matrix theory, and threshold phenomena are presented. The book offers a self-contained introduction to concentration inequalities, including a survey of concentration of sums of independent random variables, variance bounds, the entropy method, and the transportation method. Deep connections with isoperimetric problems are revealed. Special attention is paid to applications to the supremum of empirical processes.

*Gopinath Kallianpur and P Sundar*

- Published in print:
- 2014
- Published Online:
- April 2014
- ISBN:
- 9780199657063
- eISBN:
- 9780191781759
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199657063.001.0001
- Subject:
- Mathematics, Probability / Statistics, Applied Mathematics

Starting with the construction of stochastic processes, the book introduces Brownian motion and martingales. After proving the Doob-Meyer decomposition, quadratic variation processes and local ...
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Starting with the construction of stochastic processes, the book introduces Brownian motion and martingales. After proving the Doob-Meyer decomposition, quadratic variation processes and local martingales are discussed. The book proceeds to construct stochastic integrals, prove the Itô formula, derive several important applications of the formula such as the martingale representation theorem and the Burkhölder-Davis-Gundy inequality, and establish the Girsanov theorem on change of measures. Next, attention is focused on stochastic differential equations which arise in modeling physical phenomena, perturbed by random forces. Diffusion processes are solutions of stochastic differential equations and form the main theme of this book. After establishing the existence and uniqueness of strong solutions to stochastic differential equations, weak solutions and martingale problems posed by stochastic differential equations are studied in detail. The Stroock-Varadhan martingale problem is a powerful tool in solving stochastic differential equations and is discussed in a separate chapter. The connection between diffusion processes and partial differential equations is quite important and fruitful. Probabilistic representations of solutions of partial differential equations, and a derivation of the Kolmogorov forward and backward equations are provided. Gaussian solutions of stochastic differential equations, and Markov processes with jumps are presented in successive chapters. The final objective of the book consists in giving a careful treatment of the probabilistic behavior of diffusions such as existence and uniqueness of invariant measures, ergodic behavior, and large deviations principle in the presence of small noise.Less

Starting with the construction of stochastic processes, the book introduces Brownian motion and martingales. After proving the Doob-Meyer decomposition, quadratic variation processes and local martingales are discussed. The book proceeds to construct stochastic integrals, prove the Itô formula, derive several important applications of the formula such as the martingale representation theorem and the Burkhölder-Davis-Gundy inequality, and establish the Girsanov theorem on change of measures. Next, attention is focused on stochastic differential equations which arise in modeling physical phenomena, perturbed by random forces. Diffusion processes are solutions of stochastic differential equations and form the main theme of this book. After establishing the existence and uniqueness of strong solutions to stochastic differential equations, weak solutions and martingale problems posed by stochastic differential equations are studied in detail. The Stroock-Varadhan martingale problem is a powerful tool in solving stochastic differential equations and is discussed in a separate chapter. The connection between diffusion processes and partial differential equations is quite important and fruitful. Probabilistic representations of solutions of partial differential equations, and a derivation of the Kolmogorov forward and backward equations are provided. Gaussian solutions of stochastic differential equations, and Markov processes with jumps are presented in successive chapters. The final objective of the book consists in giving a careful treatment of the probabilistic behavior of diffusions such as existence and uniqueness of invariant measures, ergodic behavior, and large deviations principle in the presence of small noise.