Advances In Stochastic Inequalities

Advances in Stochastic Inequalities

How to Gamble If You Must

This classic of advanced statistics is geared toward graduate-level readers and uses the concepts of gambling to develop important ideas in probability theory. The authors have distilled the essence of many years' research into a dozen concise chapters. "Strongly recommended" by the Journal of the American Statistical Association upon its initial publication, this revised and updated edition features contributions from two well-known statisticians that include a new Preface, updated references, and findings from recent research. Following an introductory chapter, the book formulates the gambler's problem and discusses gambling strategies. Succeeding chapters explore the properties associated with casinos and certain measures of subfairness. Concluding chapters relate the scope of the gambler's problems to more general mathematical ideas, including dynamic programming, Bayesian statistics, and stochastic processes. Dover (2014) revised and updated republication of the 1976 Dover edition entitled Inequalities for Stochastic Processes. See every Dover book in print at www.doverpublications.com

Probabilistic Inequalities

1. Introduction -- 2. Basic stochastic Ostrowski inequalities -- 3. Multidimensional Montgomery identities and Ostrowski type inequalities -- 4. General probabilistic inequalities -- 5. About Grothendieck inequalities -- 6. Basic optimal estimation of Csiszar's f-divergence -- 7. Approximation via representations of Csiszar's f-divergence -- 8. Sharp high degree estimation of Csiszar's f-divergence -- 9. Csiszar's f-divergence as a measure of dependence --10. Optimal estimation of discrete Csiszar f-divergence -- 11. About a general discrete measure of dependence -- 12. Hölder-Like Csiszar's f-divergence inequalities -- 13. Csiszar's discrimination and Ostrowski inequalities via Euler-type and Fink identities -- 14. Taylor-Widder representations and Grüss, Means, Ostrowski and Csiszar's inequalities -- 15. Representations of functions and Csiszar's f-divergence -- 16. About general moment theory -- 17. Extreme bounds on the average of a rounded off observation under a moment condition -- 18. Moment theory of random rounding rules subject to one moment condition -- 19. Moment theory on random rounding rules using two moment conditions -- 20. Prokhorov radius around zero using three moment constraints -- 21. Precise rates of Prokhorov convergence using three moment conditions -- 22. On Prokhorov convergence or probability measures to the unit under three moments -- 23. Geometric moment methods applied to optimal portfolio -- 24. Discrepancies between general integral means -- 25. Grüss type inequalities using the Stieltjes integral -- 26. Chebyshev-Grüss type and difference of integral means inequalities using the Stieltjes integral -- 27. An expansion formula -- 28. Integration by parts on the multidimensional domain.