Concentration inequalities, which express the fact that certain complicated random variables are almost constant, have proven of utmost importance in many areas of probability and statistics. This volume contains refined versions of these inequalities, and their relationship to many applications particularly in stochastic analysis. The broad range and the high quality of the contributions make this book highly attractive for graduates, postgraduates and researchers in the above areas.
Concentration inequalities, which express the fact that certain complicated random variables are almost constant, have proven of utmost importance in many areas of probability and statistics. This volume contains refined versions of these inequalities, and their relationship to many applications particularly in stochastic analysis. The broad range and the high quality of the contributions make this book highly attractive for graduates, postgraduates and researchers in the above areas.
I. Geometric Inequalities.- Large Deviations of Typical Linear Functionals on a Convex Body with Unconditional Basis.- A Concentration Inequality on Riemannian Path Space.- A Remark on Unified Error Exponents: Hypothesis Testing, Data Compression and Measure Concentration.- Concentration Inequalities for Convex Functions on Product Spaces.- II. Independent Random Vectors, Chaos, Martingales and Lévy Processes.- Exponential Inequalities, with Constants, for U-statistics of Order Two.- On a.s. Unconditional Convergence of Random Series in Banach Spaces.- Moment and Tail Estimates for Multidimensional Chaoses Generated by Positive Random Variables with Logarithmically Concave Tails.- A Quantitative Law of Large Numbers via Exponential Martingales.- Sufficient Conditions for Boundedness of Moving Average Processes.- Notes on the Speed of Entropic Convergence in the Central Limit Theorem.- On a Nonsymmetric Version of the Khinchine-Kahane Inequality.- Dimensionality Reduction in Extremal Problems for Moments of Linear Combinations of Vectors with Random Coefficients.- III. Empirical Processes.- Moderate Deviations of Empirical Processes.- Concentration Inequalities for Sub-Additive Functions Using the Entropy Method.- Ratio Limit Theorems for Empirical Processes.- Asymptotic Distributions of Trimmed Wasserstein Distances Between the True and the Empirical Distribution Functions.- IV. Stochastic Differential Equations.- On the Rate of Convergence of Splitting-up Approximations for SPDEs.- Lower Bounds for Densities of Uniformly Elliptic Non-homogeneous Diffusions.- Lyapunov Exponents of Nonlinear Stochastic Differential Equations with Jumps.- Stochastic Differential Equations with Additive Fractional Noise and Locally Unbounded Drift.
I. Geometric Inequalities.- Large Deviations of Typical Linear Functionals on a Convex Body with Unconditional Basis.- A Concentration Inequality on Riemannian Path Space.- A Remark on Unified Error Exponents: Hypothesis Testing, Data Compression and Measure Concentration.- Concentration Inequalities for Convex Functions on Product Spaces.- II. Independent Random Vectors, Chaos, Martingales and Lévy Processes.- Exponential Inequalities, with Constants, for U-statistics of Order Two.- On a.s. Unconditional Convergence of Random Series in Banach Spaces.- Moment and Tail Estimates for Multidimensional Chaoses Generated by Positive Random Variables with Logarithmically Concave Tails.- A Quantitative Law of Large Numbers via Exponential Martingales.- Sufficient Conditions for Boundedness of Moving Average Processes.- Notes on the Speed of Entropic Convergence in the Central Limit Theorem.- On a Nonsymmetric Version of the Khinchine-Kahane Inequality.- Dimensionality Reduction in Extremal Problems for Moments of Linear Combinations of Vectors with Random Coefficients.- III. Empirical Processes.- Moderate Deviations of Empirical Processes.- Concentration Inequalities for Sub-Additive Functions Using the Entropy Method.- Ratio Limit Theorems for Empirical Processes.- Asymptotic Distributions of Trimmed Wasserstein Distances Between the True and the Empirical Distribution Functions.- IV. Stochastic Differential Equations.- On the Rate of Convergence of Splitting-up Approximations for SPDEs.- Lower Bounds for Densities of Uniformly Elliptic Non-homogeneous Diffusions.- Lyapunov Exponents of Nonlinear Stochastic Differential Equations with Jumps.- Stochastic Differential Equations with Additive Fractional Noise and Locally Unbounded Drift.
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