Concentration and Gaussian Approximation for Randomized Sums

• Part I. Generalities.- 1. Moments and correlation conditions.- 2. Some classes of probability distributions.- 3. Characteristic functions.- 4. Sums of independent random variables.- Part II. Selected topics on concentration.- 5. Standard analytic conditions.- 6. Poincaré-type inequalities.- 7. Logarithmic Sobolev inequalities.- 8. Supremum and infimum convolutions.- Part IV. Analysis on the sphere.- 9. Sobolev-type inequalities.- 10. Second order spherical concentration.- 11. Linear functionals on the sphere.- Part V. First applications to randomized sums.- 12. Typical distributions.- 13. Characteristic functions of weighted sums.- 14. Fluctuations of distributions.- Part VI. Refined bounds and rates.- 15. L^2 expansions and estimates.- 16. Refinements for the Kolmogorov distance.- 17. Applications of the second order correlation condition.- Part VII. Distributions and coefficients of special types.- 18. Special systems and examples.- 19. Distributions with symmetries.- 20. Productmeasures.- 21. Coefficients of Special type.- Glossary.

“The exposition in the book is careful and thorough, with proofs given throughout. Each chapter concludes with remarks to give context, references and recent developments in the topic at hand, making this an excellent starting point for a study of the randomised sums considered here. While they begin with fundamental results in the area, the authors move carefully and efficiently to state-of-the-art techniques and results. Overall, this is a well-written book that will prove invaluable to researchers in the area.” (Fraser Daly, zbMATH 1542.60001, 2024)“The book under review is a much-anticipated item by researchers, especially because it focuses on dependent random variables. It is worthy of recommendation to all those dealing with limit theorems for independent and dependent random variables and those studying the properties of multivariate random vectors.” (Przemyslaw Matula, Mathematical Reviews, November, 2024)