Introduction to Probability :`Theory and its Applications Vol - l

Contents
The Exponential and the Uniform Densities
1
Densities Convolutions
3
The Exponential Density
8
Waiting Time Paradoxes The Poisson Process
11
The Persistence of Bad Luck
15
Waiting Times and Order Statistics
17
The Uniform Distribution
21
Random Splittings
25
Terminating Transient Processes
374
Diverse Applications
377
Existence of Limits in Stochastic Processes
379
9 Renewal Theory on the Whole Line
380
Problems for Solution
385
Random Walks in 311
389
Basic Concepts and Notations
390
Duality Types of Random Walks
394

Convolutions and Covering Theorems
26
Random Directions
29
The Use of Lebesgue Measure
33
Empirical Distributions
36
Problems for Solution
39
Special Densities Randomization
45
Gamma Distributions
47
3 Related Distributions of Statistics
48
Some Common Densities
49
Randomization and Mixtures
53
Discrete Distributions
55
CHAPTER
66
chapter
103
CHAPTER
127
Symmetrization
148
Integration by Parts Existence of Moments
150
Chebyshevs Inequality
151
Further Inequalities Convex Functions
152
Simple Conditional Distributions Mixtures
156
Starred sections are not required for the understanding of the sequel and should be omitted at first readme
160
11 Conditional Expectations
162
Problems for Solution
165
A Survey of some Important Distributions and Processes
169
Examples
173
Infinitely Divisible Distributions in Rl
176
Processes with Independent Increments
179
5 Ruin Problems in Compound Poisson Processes
182
Renewal Processes
184
Examples and Problems
187
Random Walks
190
The Queuing Process
194
Persistent and Transient Random Walks
200
General Markov Chains
205
12 Martingales
209
Problems for Solution
215
Laws of Large Numbers Applications in Analysis
219
Bernstein Polynomials Absolutely Monotone Functions
222
Moment Problems
224
4 Application to Exchangeable Variables
228
5 Generalized Taylor Formula and SemiGroups
230
Inversion Formulas for Laplace Transforms
232
7 Laws of Large Numbers for Identically Distributed Variables
234
8 Strong Laws
237
9 Generalization to Martingales
241
Problems for Solution
244
The Basic Limit Theorems
247
Special Properties
252
Distributions as Operators
254
The Central Limit Theorem
258
5 Infinite Convolutions
265
Selection Theorems
267
7 Ergodic Theorems for Markov Chains
270
Regular Variation
275
9 Asymptotic Properties of Regularly Varying Functions
279
Problems for Solution
284
chapter K Infinitely Divisible Distributions and SemiGroups
290
Convolution SemiGroups
293
Preparatory Lemmas
296
Finite Variances
298
The Main Theorems
300
Stable SemiGroups
305
Triangular Arrays with Identical Distributions
308
Domains of Attraction
312
Variable Distributions The ThreeSeries Theorem
316
Problems for Solution
318
Markov Processes and SemiGroups
321
The PseudoPoisson Type
322
Linear Increments
324
Jump Processes
326
Diffusion Processes in 311
332
The Forward Equation Boundary Conditions
337
Diffusion in Higher Dimensions
344
Subordinated Processes
345
Markov Processes and SemiGroups
349
The Exponential Formula of SemiGroup Theory
353
Generators The Backward Equation
356
Renewal Theory
358
Proof of the Renewal Theorem
364
3 Refinements
366
Persistent Renewal Processes
368
The Number Nt of Renewal Epochs
372
Distribution of Ladder Heights WienerHopf Factor ization
398
3a The WienerHopf Integral Equation
402
Examples
404
Applications
408
A Combinatorial Lemma
412
Distribution of Ladder Epochs
413
The Arc Sine Laws
417
Miscellaneous Complements
423
Problems for Solution
425
Laplace Transforms Tauberian Theorems Resolvents
429
Elementary Properties
434
Examples
436
Completely Monotone Functions Inversion Formulas
439
Tauberian Theorems
442
6 Stable Distributions
448
7 Infinitely Divisible Distributions
449
8 Higher Dimensions
452
Laplace Transforms for SemiGroups
454
The HilleYosida Theorem
459
Problems for Solution
463
Applications of Laplace Transforms
466
Examples
468
Limit Theorems Involving Arc Sine Distributions
470
Busy Periods and Related Branching Processes
473
Diffusion Processes
475
BirthandDeath Processes and Random Walks
479
The Kolmogorov Differential Equations
483
The Pure Birth Process
488
Calculation of Ergodic Limits and of FirstPassage Times
491
Problems for Solution
495
Characteristic Functions
498
Special Distributions Mixtures
502
2a Some Unexpected Phenomena
505
Uniqueness Inversion Formulas
507
Regularity Properties
511
The Central Limit Theorem for Equal Components
515
The Lindeberg Conditions
518
Characteristic Functions in Higher Dimensions
521
8 Two Characterizations of the Normal Distribution
525
Problems for Solution
526
CHAPTER XVI Expansions Related to the Central Limit Theorem
531
Notations
532
Expansions for Densities
533
Smoothing
536
Expansions for Distributions
538
The BerryEsseen Theorems
543
Expansions in the Case of Varying Components
546
Large Deviations
549
Infinitely Divisible Distributions
554
Canonical Forms The Main Limit Theorem
558
2a Derivatives of Characteristic Functions
565
Examples and Special Properties
566
Special Properties
570
Stable Distributions and Their Domains of Attraction
574
6 Stable Densities
581
Triangular Arrays
583
8 The Class L
588
9 Partial Attraction Universal Laws
590
10 Infinite Convolutions
592
Higher Dimensions
593
Problems for Solution
595
Applications of Fourier Methods to Random Walks
598
2 Finite Intervals Walds Approximation
601
The WienerHopf Factorization
604
Implications and Applications
609
Two Deeper Theorems
612
Criteria for Persistency
614
Problems for Solution
616
Harmonic Analysis
619
Positive Definite Functions
620
Stationary Processes
623
Fourier Series
626
5 The Poisson Summation Formula
629
Positive Definite Sequences
633
L2 Theory
635
Stochastic Processes and Integrals
641
Problems for Solution
647
Answers to Problems
651
Some Books on Cognate Subjects
655
Index
657
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