Odile Pons Boeken

Focusing on classical inequalities in vector and functional spaces, this book explores their applications in probability theory. It presents innovative analytical inequalities that provide improved bounds and generalizations for various mathematical concepts, including the sum and supremum of random variables, martingales, and transformed Brownian motions. Additionally, it delves into Markov processes, point processes, and various stochastic processes like renewal, branching, and shock processes, making it a valuable resource for advanced studies in probability and analysis.

Focusing on nonparametric kernel estimators, this book explores their application in statistical analysis for both independent and dependent sequences of random variables. It emphasizes the importance of bandwidth selection to minimize estimation error and demonstrates weak convergence of the estimators. Additionally, it presents new mathematical results related to density and regression functions, as well as more complex models, including point process intensity, auto-regressive diffusion parameters, and single-index regression models.

Advanced methods in integral calculus and optimization are explored, alongside classical theories of ordinary and partial differential equations. The book offers explicit solutions for both linear and nonlinear differential equations, as well as implicit solutions featuring discrete approximations, making it a comprehensive resource for those studying advanced mathematical concepts.

Focusing on advanced statistical methods, this book explores the approximation and estimation of nonparametric functions through projections onto orthonormal function bases. It introduces series estimators that enhance density estimators for various complex models, including mixture, deconvolution, and semi-parametric models. The authors demonstrate optimal convergence rates in Hilbert spaces and analyze mean square errors relative to basis size, utilizing cross-validation for consistent estimation. Additionally, wavelet estimators are examined within these frameworks, providing a comprehensive approach to modern data analysis techniques.