Csörgo / Révész / Birnbaum Strong Approximations in Probability and Statistics

1. Auflage 2014
ISBN: 978-1-4832-6804-0
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 286 Seiten, Web PDF

Strong Approximations in Probability and Statistics presents 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 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 demonstrate the validity of previously discussed theorems, including Brownian bridges and Kiefer process, for empirical and quantile processes. Chapter 6 illustrate the approximation of defined sequences of empirical density, regression, and characteristic functions by appropriate Gaussian processes. Chapter 7 deal 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 advance mathematics students.

Autoren/Hrsg.

Csörgo, M.

Révész, P.

Birnbaum, Z. W.

Lukacs, E.

Inhaltsverzeichnis

1;Front Cover;1
2;Strong Approximations in Probability and Statistics;4
3;Copyright Page;5
4;Table of Contents;6
5;Preface;10
6;Introduction;12
7;Chapter 1. Wiener and some Related Gaussian Processes;22
7.1;1.0 On the notion of a Wiener process;22
7.2;1.1 Definition and existence of a Wiener process;22
7.3;1.2 How big are the increments of a Wiener process?;30
7.4;1.3 The law of iterated logarithm for the Wiener process;37
7.5;1.4 Brownian bridges;42
7.6;1.5 The distributions of some functional of the Wiener and Brownian bridge processes;44
7.7;1.6 The modulus of non-differentiability of the Wiener process;45
7.8;1.7 How small are the increments of a Wiener process?;48
7.9;1.8 Infinite series representations of the Wiener process and Brownian bridge;54
7.10;1.9 The Ornstein-Uhlenbeck process;56
7.11;1.10 On the notion of a two-parameter Wiener process;58
7.12;1.11 Definition and existence of a two-parameter Wiener process;59
7.13;1.12 How big are the increments of a two-parameter Wiener process?;62
7.14;1.13 A continuity modulus of W(x, y);75
7.15;1.14 The limit points of W(x, y) as y–8;76
7.16;1.15 The Kiefer process;81
7.17;Supplementary remarks;83
8;Chapter 2. Strong Approximations of Partial Sums of I. I.D.R.V. by Wiener Processes;89
8.1;2.0 Notations;89
8.2;2.1 A proof of Donsker's theorem with Skorohod's embedding scheme;89
8.3;2.2 The strong invariance principle appears;92
8.4;2.3 The stochastic Geyser problem as a lower limit to the strong invariance problem;96
8.5;2.4 The longest runs of pure heads and the stochastic Geyser problem;98
8.6;2.5 Improving the upper limit;102
8.7;2.6 The best rates emerge;107
8.8;Supplementary remarks;113
9;Chapter 3. A Study of Partial Sums with the Help of Strong Approximation Methods;116
9.1;3.0 Introduction;116
9.2;3.1 How big are the increments of partial sums of I.I.D.R.V. when the moment generating function exists?;116
9.3;3.2 How big are the increments of partial sums of I.I.D.R.V. when the moment generating function does not exist?;118
9.4;3.3 How small are the increments of partial sums of I.I.D.R.V.?;121
9.5;3.4 A summary;123
9.6;Supplementary remarks;123
10;Chapter 4. Strong Approximations of Empirical Processes by Gaussian Processes;128
10.1;4.1 Some classical results;128
10.2;4.2 Why should the empirical process behave like a Brownian bridge?;130
10.3;4.3 The first strong approximations of the empirical process;131
10.4;4.4 Best strong approximations of the empirical process;134
10.5;4.5 Strong approximation of the quantile process;144
10.6;Supplementary remarks;155
11;Chapter 5. A Study of Empirical and Quantile Processes with the Help of Strong Approximation Methods;157
11.1;5.0 Introduction;157
11.2;5.1 The law of iterated logarithm for the empirical process;157
11.3;5.2 The distance between the empirical and the quantile processes;161
11.4;5.3 The law of iterated logarithm for the quantile process;163
11.5;5.4 Asymptotic distribution results for some classical functionals of the empirical process;164
11.6;5.5 Asymptotic distribution results for some classical functionals of the quantile process;172
11.7;5.6 Asymptotic distribution results for some classical functionals of some k-sample empirical and quantile processes;182
11.8;5.7 Approximations of the empirical process when parameters are estimated;189
11.9;5.8 Asymptotic quadratic quantile tests for composite goodness-of-fit;203
11.10;5.9 On testing for exponentiality;214
11.11;Supplementary remarks;217
12;Chapter 6. A Study of Further Empirical Processes with the Help of Strong Approximation Methods;220
12.1;6.0 Introduction;220
12.2;6.1 Strong invariance principles and limit distributions for empirical densities;220
12.3;6.2 Strong theorems for empirical densities;231
12.4;6.3 Empirical regression;238
12.5;6.4 Empirical characteristic functions;243
12.6;Supplementary remarks;249
13;Chapter 7. Random Limit Theorems via Strong Invariance Principles;251
13.1;7.0 Introduction and some historical remarks;251
13.2;7.1 Laws of large numbers for randomly selected sequences;253
13.3;7.2 Invariance (strong and weak) principles for random-sum limit theorems;256
13.4;7.3 Invariance (strong and weak) principles for random size empirical processes;262
14;References;264
15;Author Index;278
16;Subject Index;282
17;Summary of Notations and Abbreviations;284