Karlin A First Course in Stochastic Processes

1. Auflage 2014
ISBN: 978-1-4832-6809-5
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 514 Seiten, Web PDF

A First Course in Stochastic Processes focuses on several principal areas of stochastic processes and the diversity of applications of stochastic processes, including Markov chains, Brownian motion, and Poisson processes. The publication first takes a look at the elements of stochastic processes, Markov chains, and the basic limit theorem of Markov chains and applications. Discussions focus on criteria for recurrence, absorption probabilities, discrete renewal equation, classification of states of a Markov chain, and review of basic terminologies and properties of random variables and distribution functions. The text then examines algebraic methods in Markov chains and ratio theorems of transition probabilities and applications. The manuscript elaborates on the sums of independent random variables as a Markov chain, classical examples of continuous time Markov chains, and continuous time Markov chains. Topics include differentiability properties of transition probabilities, birth and death processes with absorbing states, general pure birth processes and Poisson processes, and recurrence properties of sums of independent random variables. The book then ponders on Brownian motion, compounding stochastic processes, and deterministic and stochastic genetic and ecological processes. The publication is a valuable source of information for readers interested in stochastic processes.

Autoren/Hrsg.

Karlin, Samuel

Inhaltsverzeichnis

1;Front Cover;1
2;A First Course in Stochastic Processes;4
3;Copyright Page;5
4;Table of Contents;8
5;Preface;6
6;Chapter 1. ELEMENTS OF STOCHASTIC PROCESSES;14
6.1;1. Review of Basic Terminology and Properties of Random Variables and Distribu tion Functions;14
6.2;2. Two Simple Examples of Stochastic Processes;24
6.3;3. Classification of General Stochastic Processes;29
6.4;Problems;34
6.5;References;39
7;Chapter 2. MARKOV CHAINS;40
7.1;1. Definitions;40
7.2;2. Examples of Markov Chains;42
7.3;3. Transition Probability Matrices of a Markov Chain;53
7.4;4. Classification of States of a Markov Chain;54
7.5;5. Recurrence;57
7.6;6. Examples of Recurrent Markov Chains;62
7.7;7. More on Recurrence;67
7.8;Problems;68
7.9;References;73
8;Chapter 3. THE BASIC LIMIT THEOREM OF MARKOV CHAINS AND APPLICATIONS;74
8.1;1. Discrete Renewal Equation;74
8.2;2. Proof of Theorem 1.1;80
8.3;3. Absorption Probabilities;82
8.4;4. Criteria for Recurrence;87
8.5;5. A Queueing Example;89
8.6;6. Another Queueing Model;95
8.7;7. Random Walk;99
8.8;Problems;101
8.9;References;106
9;Chapter 4. ALGEBRAIC METHODS IN MARKOV CHAINS;107
9.1;1. Preliminaries;107
9.2;2. Relations of Eigenvalues and Recurrence Classes;109
9.3;3. Periodic Classes;113
9.4;4. Special Computational Methods in Markov Chains;116
9.5;5. Examples;120
9.6;6. Applications to Coin Tossing;125
9.7;Problems;130
9.8;References;137
10;Chapter 5. RATIO THEOREMS OF TRANSITION PROBABILITIES AND APPLICATIONS;138
10.1;1. Taboo Probabilities;138
10.2;2. Ratio Theorems;140
10.3;3. Existence of Generalized Stationary Distributions;145
10.4;4. Interpretation of Generalized Stationary Distributions;149
10.5;5. Regular, Superregular, and Subregular Sequences for Markov Chains;152
10.6;Problems;158
10.7;References;161
11;Chapter 6. SUMS OF INDEPENDENT RANDOM VARIABLES AS A MARKOV CHAIN;162
11.1;1. Recurrence Properties of Sums of Independent Random Variables;162
11.2;2. Local Limit Theorems;166
11.3;3. Right Regular Sequences for the Markov Chain {Sn};173
11.4;Problems;183
11.5;References;187
12;Chapter 7. CLASSICAL EXAMPLES OF CONTINUOUS TIME MARKOV CHAINS;188
12.1;1. General Pure Birth Processes and Poisson Processes;188
12.2;2. More about Poisson Processes;194
12.3;3. A Counter Model;198
12.4;4. Birth and Death Processes;202
12.5;5. Differential Equations of Birth and Death Processes;205
12.6;6. Examples of Birth and Death Processes;208
12.7;7. Birth and Death Processes with Absorbing States;214
12.8;8. Finite State Continuous Time Markov Chains;219
12.9;Problems;221
12.10;References;230
13;Chapter 8. CONTINUOUS TIME MARKOV CHAINS;231
13.1;1. Differentiability Properties of Transition Probabilities;231
13.2;2. Conservative Processes and the Forward and Backward Differential Equations;236
13.3;3. Construction of a Continuous Time Markov Chain from Its Infinitesimal Parameters;238
13.4;4. Strong Markov Property;243
13.5;Problems;246
13.6;References;248
14;Chapter 9. ORDER STATISTICS, POISSON PROCESSES, AND APPLICATIONS;249
14.1;1. Order Statistics and Their Relation to Poisson Processes;249
14.2;2. The Ballot Problem;257
14.3;3. Empirical Distribution Functions and Order Statistics;263
14.4;4. Some Limit Distributions for Empirical Distribution Functions;269
14.5;Problems;274
14.6;References;283
15;Chapter 10. BROWNIAN MOTION;284
15.1;1. Background Material;284
15.2;2. Joint Probabihties for Brownian Motion;286
15.3;3. Continuity of Paths and the Maximum Variables;289
15.4;Problems;293
15.5;References;298
16;Chapter 11. BRANCHING PROCESSES;299
16.1;1. Discrete Time Branching Processes;299
16.2;2. Generating Function Relations for Branching Processes;301
16.3;3. Extinction Probabilities;303
16.4;4. Examples;307
16.5;5. Two-Type Branching Processes;311
16.6;6. Multi-Type Branching Processes;318
16.7;7. Continuous Time Branching Processes;319
16.8;8. Extinction Probabihties for Continuous Time Branching Processes;323
16.9;9. Limit Theorems for Continuous Time Branching Processes;326
16.10;10. Two-Type Continuous Time Branching Process;331
16.11;11. Branching Processes with General Variable Lifetime;338
16.12;Problems;343
16.13;References;348
17;Chapter 12. COMPOUNDING STOCHASTIC PROCESSES;349
17.1;1. Multidimensional Homogeneous Poisson Processes;350
17.2;2. An Apphcation of Multidimensional Poisson Processes to Astronomy;357
17.3;3. Immigration and Population Growth;358
17.4;4. Stochastic Models of Mutation and Growth;361
17.5;5. One-Dimensional Geometric Population Growth;366
17.6;6. Stochastic Population Growth Model in Space and Time;369
17.7;7. Deterministic Population Growth with Age Distribution;373
17.8;8. A Discrete Aging Model;379
17.9;Problems;380
17.10;References;385
18;Chapter 13. DETERMINISTIC AND STOCHASTIC GENETO AND ECOLOGICAL PROCESSES;386
18.1;1. Genetic Models; Description of the Genetic Mechanism;386
18.2;2. Inbreeding;395
18.3;3. Polyploidy;401
18.4;4. Markov Processes Induced by Direct Product Branching Processes;403
18.5;5. Multi-Type Population Frequency Models;409
18.6;6. Eigenvalues of Markov Chains Induced by Direct Product Branching Processses;412
18.7;7. Eigenvalues of Multi-Type Mutation Model;420
18.8;8. Probabihstic Interpretations of the Eigenvalues;430
18.9;Problems;438
18.10;References;442
19;Chapter 14. QUEUEING PROCESSES;443
19.1;1. General Description;443
19.2;2. The Simplest Queueing Processes (M/M/1);444
19.3;3. Some General One-Server Queueing Models;446
19.4;4. Embedded Markov Chain Method Apphed to the Queueing Model (M/GI/1);452
19.5;5. Exponential Service Times (G/M/1);458
19.6;6. Gamma Arrival Distribution and Generalizations
(Ek/M/1);461
19.7;7: Exponential Service with s Servers (GI/M/s);466
19.8;8: The Virtual Waiting Time and the Busy Period;469
19.9;Problem;475
19.10;REFERENCES;481
20;APPENDIX.
REVIEW OF MATRIX ANALYSIS;482
20.1;1 : The Spectral Theorem;482
20.2;2: The Frobenius Theory of Positive Matrices;488
21;MISCELLANEOUS
PROBLEMS;498
22;INDEX;512