Anderson / Athreya / Iglehart Probability, Statistics, and Mathematics

Papers in Honor of Samuel Karlin
1. Auflage 2014
ISBN: 978-1-4832-1600-3
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 412 Seiten, Web PDF

Probability, Statistics, and Mathematics: Papers in Honor of Samuel Karlin is a collection of papers dealing with probability, statistics, and mathematics. Conceived in honor of Polish-born mathematician Samuel Karlin, the book covers a wide array of topics, from the second-order moments of a stationary Markov chain to the exponentiality of the local time at hitting times for reflecting diffusions. Smoothed limit theorems for equilibrium processes are also discussed. Comprised of 24 chapters, this book begins with an introduction to the second-order moments of a stationary Markov chain, paying particular attention to the consequences of the autoregressive structure of the vector-valued process and how to estimate the stationary probabilities from a finite sequence of observations. Subsequent chapters focus on A. Selberg's second beta integral and an integral of mehta; a normal approximation for the number of local maxima of a random function on a graph; nonnegative polynomials on polyhedra; and the fundamental period of the queue with Markov-modulated arrivals. The rate of escape problem for a class of random walks is also considered. This monograph is intended for students and practitioners in the fields of statistics, mathematics, and economics.

Autoren/Hrsg.

Anderson, T. W.

Athreya, Krishna B.

Iglehart, Donald L.

Inhaltsverzeichnis

1;Front Cover;1
2;Probability, Statistics, and Mathematics: Papers in Honor of Samuel Karlin;4
3;Copyright Page;5
4;Table of Contents;6
5;LIST OF CONTRIBUTORS;10
6;FOREWORD;12
7;PUBLICATIONS OF SAMUEL KARLIN;14
8;Chapter 1. Second-Order Moments of a Stationary Markov Chain and Some Applications;42
8.1;1. INTRODUCTION;42
8.2;2. A STATIONARY MARKOV CHAIN;43
8.3;3. SECOND-ORDER MOMENTS;44
8.4;4. ESTIMATION OF THE STATIONARY PROBABILITIES;48
8.5;5. SCORING A MARKOV CHAIN;53
8.6;REFERENCES;56
9;Chapter 2. A "Dynamic" Proof of the Frobenius-Perron Theorem for Metzler Matrices;58
9.1;APPENDIX;64
9.2;REFERENCES;66
10;Chapter 3. Selberg's Second Beta Integral and an Integral of Mehta;68
10.1;1. INTRODUCTION;68
10.2;2. SELBERG'S GAMMA INTEGRAL via AOMOTO'S ARGUMENT;69
10.3;3. SELBERG'S SECOND BETA INTEGRAL;73
10.4;4. SOME INTEGRALS OF MEHTA;75
10.5;REFERENCES;79
11;Chapter 4. Exponentiality of the Local Time at Hitting Times for Reflecting Diffusions and an Application;82
11.1;1. INTRODUCTION;82
11.2;2. REFLECTING BROWNIAN MOTION;83
11.3;3. REFLECTING DIFFUSIONS;86
11.4;4. GENERAL INITIAL CONDITIONS;90
11.5;5. THE PROCESS;91
11.6;6. AN APPLICATION TO A STOCHASTIC CONTROL PROBLEM;93
11.7;APPENDIX;97
11.8;REFERENCES;99
12;Chapter 5. A Normal Approximation for the Number of Local Maxima of a Random Function on a Graph;100
12.1;1. INTRODUCTION;100
12.2;2. A NORMAL APPROXIMATION THEOREM FOR SUMS OF INDICATOR RANDOM VARIABLES;102
12.3;3. A NORMAL APPROXIMATION FOR THE DISTRIBUTION OF THE NUMBER OF LOCAL MAXIMA;107
12.4;4. EXAMPLES;120
12.5;REFERENCES;122
13;Chapter 6. Operator Solution of Infinite Gd Games of Imperfect Information;124
13.1;1. INTRODUCTION;124
13.2;2. THE OPERATOR;125
13.3;3. THE RESULT;125
13.4;4. COMPLEXITY OF THE PROBLEM;127
13.5;REFERENCES;128
14;Chapter 7. Smoothed Limit Theorems for Equilibrium Processes;130
14.1;1. INTRODUCTION;130
14.2;2. THE TWO BASIC FORMS OF THE RENEWAL THEOREM;131
14.3;3. LIMIT THEOREMS FOR EQUILIBRIUM PROCESSES;134
14.4;4. A SMOOTHED VERSION OF THE RENEWAL THEOREM;138
14.5;5. SMOOTHED LIMIT THEOREMS FOR EQUILIBRIUM PROCESSES;141
14.6;REFERENCES;143
15;Chapter 8. Supercritical Branching Processes with Countably Many Types and the Size of Random Cantor Sets;144
15.1;1. INTRODUCTION;144
15.2;2. PROOF OF THEOREM 1;154
15.3;REFERENCES;162
16;Chapter 9. Maxima of Random Quadratic Forms on a Simplex;164
16.1;1. AN OLD PROBLEM IN POPULATION GENETICS;164
16.2;2. THE PROBABILITY OF STABILITY;166
16.3;3. THE ASYMPTOTIC FORM OF Snr(A);168
16.4;4. THE FUNDAMENTAL CONE;172
16.5;5. THE SHAPE OF A TYPICAL POLYMORPHISM;175
16.6;6. A LOWER BOUND FOR GENERAL r;178
16.7;REFERENCES;181
17;Chapter 10. Total Positivity and Renewal Theory;182
17.1;1. INTRODUCTION;183
17.2;2. THE TOTAL POSITIVITY CONNECTION;189
17.3;3. THE CONVERGENCE THEOREM;193
17.4;4. CONTINUOUS TIME;199
17.5;REFERENCES;202
18;Chapter 11. Some Remarks on Nonnegative Polynomials on Polyhedra;204
18.1;1. INTRODUCTION;204
18.2;2. NONNEGATIVE POLYNOMIALS: PROJECTIVE COORDINATES;207
18.3;3. NONNEGATIVE BIVARIATE QUADRATIC POLYNOMIALS;217
18.4;REFERENCES;225
19;Chapter 12. The Fundamental Period of the Queue with Markov-Modulated Arrivals;228
19.1;1. INTRODUCTION;228
19.2;2. THE MMPP/G/1 QUEUE;231
19.3;3. AN EMBEDDED MARKOV PROCESS;235
19.4;ACKNOWLEDGEMENTS;239
19.5;REFERENCES;240
20;Chapter 13. Some Remarks on a Limiting Diffusion for Decomposable Branching Processes;242
20.1;1. INTRODUCTION;242
20.2;2. LIMITING DIFFUSION;244
20.3;3. RELATION BETWEEN DIFFERENT CONDITIONS;246
20.4;REFERENCES;250
21;Chapter 14. Some Results on Repeated Risktaking;252
21.1;1. INTRODUCTION;252
21.2;2. SHOULD WE BOTH GO TOGETHER WHEN WE GO?;255
21.3;3. HOW BROWNIAN CAN RATIONAL RANKING OF REPEATED GAMBLES BE?;259
21.4;4. PROPER CERTAINTY EQUIVALENTS FOR REPEATED GAMBLES;259
21.5;5. WHEN INDIFFERENCE ONCE IMPLIES INDIFFERENCE TWICE;261
21.6;6. IMPLICATIONS OF STRICT INEQUALITY IN THE LOCAL CONDITION FOR PROPERNESS;263
21.7;REFERENCES;264
22;Chapter 15. The Rate of Escape Problem for a Class of Random Walks;266
22.1;1. INTRODUCTION;266
22.2;2. PROOF OF THEOREM 2;269
22.3;3. PROOF OF THEOREM 1;273
22.4;4. FINAL REMARKS;276
22.5;REFERENCES;277
23;Chapter 16. Recent Advances on the Integrated Cauchy Functional Equation and Related Results in Applied Probability;280
23.1;1. INTRODUCTION;280
23.2;2. THEOREMS ON THE ICFE AND SOME OF ITS VARIANTS;282
23.3;3. SOME APPLICATIONS OF THE ICFE;287
23.4;REFERENCES;293
24;Chapter 17. A Note on Maximum Entropy;296
24.1;REFERENCES;300
25;Chapter 18. The Various Linear Fractional Lévy Motions;302
25.1;1. INTRODUCTION;302
25.2;2. THE VARIOUS LINEAR FRACTIONAL LÉVY MOTIONS;306
25.3;REFERENCES;310
26;Chapter 19. Bonferroni-Type Probability Bounds as an Application of the Theory of Tchebycheff Systems;312
26.1;1. INTRODUCTION;312
26.2;2. T-SYSTEMS ON;315
26.3;3. EXAMPLES;318
26.4;REFERENCES;330
27;Chapter 20. The vN Law and Repeated Risktaking;332
27.1;1. INTRODUCTION AND REVIEW;332
27.2;2. PRESENT PURPOSES;335
27.3;3. AMBIGUOUS CASES AND REASONS TO SELF-INSURE;335
27.4;4. DEFINING U(W) NEUTRAL TOWARD REPETITION OF RISKS;338
27.5;5. EXACT DERIVATION;340
27.6;6. FINAL REFLECTIONS;344
27.7;REFERENCES;346
28;Chapter 21. A Theorem in Search of a Simple Proof;348
28.1;REFERENCES;357
29;Chapter 22. Grade of Membership Representations: Concepts and Problems;358
29.1;1. INTRODUCTION;358
29.2;2. AN ELEMENTARY SPECIAL CASE;359
29.3;3. MULTIDIMENSIONAL CONTINGENCY TABLES;363
29.4;4. ESTIMATION;366
29.5;5. STOCHASTIC PROCESS;369
29.6;6. DISCUSSION;372
29.7;REFERENCES;374
30;Chapter 23. Uniform Error Bounds Involving Logspline Models;376
30.1;1. INTRODUCTION;376
30.2;2. PRELIMINARY INEQUALITIES;380
30.3;3. THE INVERSE GRAM MATRIX;382
30.4;4. LOGSPLINE MODELS;385
30.5;5. LOGSPLINE RESPONSE MODELS;389
30.6;REFERENCES;396
31;Chapter 24. An Alternative to Cp Model Selection that Emphasizes the Quality of Coefficient Estimation;398
31.1;1. INTRODUCTION;398
31.2;2. THE Rp STATISTIC;399
31.3;3. RIDGE REGRESSION;402
31.4;4. EXAMPLES;405
31.5;REFERENCES;407