Measure Theory and Probability Theory

Front Cover
Springer Science & Business Media, Jul 27, 2006 - Business & Economics - 618 pages
0 Reviews

This is a graduate level textbook on measure theory and probability theory. The book can be used as a text for a two semester sequence of courses in measure theory and probability theory, with an option to include supplemental material on stochastic processes and special topics. It is intended primarily for first year Ph.D. students in mathematics and statistics although mathematically advanced students from engineering and economics would also find the book useful. Prerequisites are kept to the minimal level of an understanding of basic real analysis concepts such as limits, continuity, differentiability, Riemann integration, and convergence of sequences and series. A review of this material is included in the appendix.

The book starts with an informal introduction that provides some heuristics into the abstract concepts of measure and integration theory, which are then rigorously developed. The first part of the book can be used for a standard real analysis course for both mathematics and statistics Ph.D. students as it provides full coverage of topics such as the construction of Lebesgue-Stieltjes measures on real line and Euclidean spaces, the basic convergence theorems, L^p spaces, signed measures, Radon-Nikodym theorem, Lebesgue's decomposition theorem and the fundamental theorem of Lebesgue integration on R, product spaces and product measures, and Fubini-Tonelli theorems. It also provides an elementary introduction to Banach and Hilbert spaces, convolutions, Fourier series and Fourier and Plancherel transforms. Thus part I would be particularly useful for students in a typical Statistics Ph.D. program if a separate course on real analysis is not a standard requirement.

Part II (chapters 6-13) provides full coverage of standard graduate level probability theory. It starts with Kolmogorov's probability model and Kolmogorov's existence theorem. It then treats thoroughly the laws of large numbers including renewal theory and ergodic theorems with applications and then weak convergence of probability distributions, characteristic functions, the Levy-Cramer continuity theorem and the central limit theorem as well as stable laws. It ends with conditional expectations and conditional probability, and an introduction to the theory of discrete time martingales.

Part III (chapters 14-18) provides a modest coverage of discrete time Markov chains with countable and general state spaces, MCMC, continuous time discrete space jump Markov processes, Brownian motion, mixing sequences, bootstrap methods, and branching processes. It could be used for a topics/seminar course or as an introduction to stochastic processes.

Krishna B. Athreya is a professor at the departments of mathematics and statistics and a Distinguished Professor in the College of Liberal Arts and Sciences at the Iowa State University. He has been a faculty member at University of Wisconsin, Madison; Indian Institute of Science, Bangalore; Cornell University; and has held visiting appointments in Scandinavia and Australia. He is a fellow of the Institute of Mathematical Statistics USA; a fellow of the Indian Academy of Sciences, Bangalore; an elected member of the International Statistical Institute; and serves on the editorial board of several journals in probability and statistics. Soumendra N. Lahiri is a professor at the department of statistics at the Iowa State University. He is a fellow of the Institute of Mathematical Statistics, a fellow of the American Statistical Association, and an elected member of the International Statistical Institute.

 

What people are saying - Write a review

We haven't found any reviews in the usual places.

Contents

III
9
IV
14
V
19
VII
25
VIII
27
IX
28
X
30
XI
31
XCII
389
XCIII
392
XCIV
393
XCV
399
XCVI
405
XCVII
417
XCVIII
424
XCIX
425

XII
39
XIII
44
XIV
47
XV
48
XVI
59
XVII
61
XVIII
71
XIX
83
XX
89
XXII
93
XXIII
94
XXIV
96
XXV
97
XXVI
98
XXVII
102
XXVIII
112
XXIX
119
XXX
125
XXXI
128
XXXII
133
XXXIII
134
XXXIV
136
XXXV
137
XXXVI
147
XXXVII
152
XXXVIII
157
XXXIX
160
XL
162
XLII
164
XLIV
166
XLV
173
XLVI
178
XLVII
181
XLVIII
189
XLIX
191
L
199
LI
212
LII
219
LIII
222
LIV
227
LV
236
LVI
240
LVII
249
LVIII
254
LIX
260
LX
262
LXI
264
LXII
266
LXIII
268
LXIV
271
LXV
274
LXVI
278
LXVII
279
LXVIII
287
LXIX
291
LXX
299
LXXI
303
LXXII
306
LXXIII
307
LXXV
309
LXXVI
316
LXXVII
323
LXXVIII
327
LXXIX
332
LXXX
337
LXXXI
343
LXXXII
352
LXXXIII
358
LXXXIV
361
LXXXVI
364
LXXXVII
368
LXXXVIII
372
LXXXIX
374
XC
376
XCI
383
CI
427
CII
429
CIII
430
CV
438
CVI
440
CVII
442
CVIII
443
CIX
457
CX
458
CXI
461
CXII
462
CXIII
464
CXIV
465
CXV
467
CXVI
469
CXVII
473
CXVIII
477
CXIX
478
CXX
480
CXXI
481
CXXII
487
CXXIII
488
CXXIV
489
CXXV
491
CXXVI
493
CXXVIII
495
CXXIX
498
CXXXI
499
CXXXII
501
CXXXIII
502
CXXXIV
503
CXXXV
504
CXXXVI
509
CXXXVII
513
CXXXVIII
514
CXXXIX
516
CXL
519
CXLI
529
CXLII
532
CXLIII
535
CXLIV
536
CXLV
537
CXLVI
540
CXLVII
545
CXLVIII
547
CXLIX
548
CL
549
CLI
552
CLII
554
CLIII
556
CLIV
561
CLV
562
CLVI
564
CLVII
566
CLVIII
568
CLIX
569
CLX
573
CLXI
574
CLXII
577
CLXIV
578
CLXV
580
CLXVI
582
CLXVII
584
CLXVIII
586
CLXIX
590
CLXX
592
CLXXII
593
CLXXIII
594
CLXXIV
599
CLXXV
600
CLXXVI
603
CLXXVII
610
CLXXVIII
612
Copyright

Other editions - View all

Common terms and phrases

References to this book