Csorgo M., Revesz P. Strong Approximations in Probability and Statistics

New York: Academic Press, 1982. — 277 p.Strong Approximations in Probability and Statistics present strong invariance type results for partial sums and empirical processes of independent and identically distributed random variables (IIDRV). This seven-chapter text emphasizes the applicability of strong approximation methodology to a variety of problems of probability and statistics.Chapter 1 evaluates the theorems for Wiener and Gaussian processes that can be extended to partial sums and empirical processes of IIDRV through strong approximation methods, while Chapter 2 addresses the problem of the best possible strong approximations of partial sums of IIDRV by a Wiener process. Chapters 3 and 4 contain theorems concerning the one-time parameter Wiener process and strong approximation for the empirical and quantile processes based on IIDRV. Chapter 5 demonstrates the validity of previously discussed theorems, including Brownian bridges and the Kiefer process, for empirical and quantile processes. Chapter 6 illustrates the approximation of defined sequences of empirical density, regression, and characteristic functions by appropriate Gaussian processes. Chapter 7 deals with the application of strong approximation methodology to study weak and strong convergence properties of random size partial sum and empirical processes.
This book will prove useful to mathematicians and advanced mathematics students.Wiener and some Related Gaussian Processes
Strong Approximations of Partial Sums of I.I.D.R.V. by Wiener Processes
A Study of Partial Sums with the Help of Strong Approximation Methods
Strong Approximations of Empirical Processes by Gaussian Processes
A Study of Empirical and Quantile Processes with the Help of Strong Approximation Methods
A Study of Further Empirical Processes with the Help of Strong Approximation Methods
Random Limit Theorems via Strong Invariance PrinciplesProbability and Mathematical Statistics: A Series of Monographs and Textbooks