An introduction to probability theory and its applications / William Feller.
Series Wiley series in probability and mathematical statistics; A Wiley publication in mathematical statisticsEditor: New York : Wiley, c1957-1971Edición: 2nd edDescripción: 2 v. ; 24 cmISBN: 0471257095 (v. 2 , v. 1 carece de ISBN)Otra clasificación: 60-01Introduction: The Nature of Probability Theory [1] 1. The Background [1] 2. Procedure [3] 3. “Statistical” Probability [4] 4. Summary [5] 5. Historical Note [6] I The Sample Space [7] 1. The Empirical Background [7] 2. Examples [9] 3. The Sample Space. Events [13] 4. Relations among Events [15] 5. Discrete Sample Spaces [17] 6. Probabilities in Discrete Sample Spaces: Preparations [19] 7. The Basic Definitions and Rules [22] 8. Problems for Solution [24] II Elements of Combinatorial Analysis [26] 1. Preliminaries [26] 2. Ordered Samples [28] 3. Examples [30] 4. Subpopulations and Partitions [32] *5. Application to Occupancy Problems [36] *5a. Application to Runs [40] 6. The Hypergeometric Distribution [41] 7. Examples for Waiting Times [45] 8. Binomial Coefficients [48] 9. Stirling’s Formula [50] Problems for Solution: 10. Exercises and Examples [53] 11. Problems and Complements of a Theoretical Character [57] 12. Problems and Identities Involving Binomial Coefficients [61] *III Fluctuations in Coin Tossing and Random Walks [65] 1. General Orientation [66] 2. Problems of Arrangements [69] 3. Random Walks and Coin Tossing [73] 4. Reformulation of the Combinatorial Theorems [74] 5. Probability of Long Leads: The First Arc Sine Law [77] 6. The Number of Returns to the Origin [81] 7. An Experimental Illustration [83] 8. Miscellaneous Complements [85] *IV Combination of Events [88] 1. Union of Events [88] 2. Application to the Classical Occupancy Problem [91] 3. The Realization of m among N Events [96] 4. Application to Matching and Guessing [97] 5. Miscellany [99] 6. Problems for Solution [101] V Conditional Probability. Stochastic Independence [104] 1. Conditional Probability [104] 2. Probabilities Defined by Conditional Probabilities. Urn Models [108] 3. Stochastic Independence [114] 4. Repeated Trials [118] *5. Applications to Genetics [121] *6. Sex-Linked Characters [125] *7. Selection [128] 8. Problems for Solution [129] VI The Binomial and the Poisson Distributions [135] 1. Bernoulli Trials [135] 2. The Binomial Distribution [136] 3. The Central Term and the Tails [139] 4. The Law of Large Numbers [141] 5. The Poisson Approximation [142] 6. The Poisson Distribution [146] 7. Observations Fitting the Poisson Distribution [149] 8. Waiting Times. The Negative Binomial Distribution [155] 9. The Multinomial Distribution [157] 10. Problems for Solution [158] VII The Normal Approximation to the Binomial Distribution [164] 1. The Normal Distribution [164] 2. The DeMoivre-Laplace Limit Theorem [168] 3. Examples [174] 4. Relation to the Poisson Approximation [176] 5. Large Deviations [178] 6. Problems for Solution [179] *VIII Unlimited Sequences of Bernoulli Trials [183] 1. Infinite Sequences of Trials [183] 2. Systems of Gambling [185] 3. The Borel-Cantelli Lemmas [188] 4. The Strong Law of Large Numbers [189] 5. The Law of the Iterated Logarithm [191] 6. Interpretation in Number Theory Language [195] 7. Problems for Solution [197] IX Random Variables; Expectation [199] 1. Random Variables [199] 2. Expectations [207] 3. Examples and Applications [209] 4. The Variance [213] 5. Covariance; Variance of a Sum [215] 6. Chebyshev’s Inequality [219] *7. Kolmogorov’s Inequality [220] *8. The Correlation Coefficient [221] 9. Problems for Solution [223] X Laws of Large Numbers [228] 1. Identically Distributed Variables [228] *2. Proof of the Law of Large Numbers [231] 3. The Theory of “Fair” Games [233] *4. The Petersburg Game [235] 5. Variable Distributions [238] *6. Applications to Combinatorial Analysis [241] *7. The Strong Law of Large Numbers [243] 8. Problems for Solution [245] XI Integral Valued Variables. Generating Functions [248] 1. Generalities [248] 2. Convolutions [250] 3. Application to First Passage and Recurrence Times in Bernoulli Trials [254] 4. Partial Fraction Expansions [257] 5. Bivariate Generating Functions [261] *6. The Continuity Theorem [262] 7. Problems for Solution [264] *XII Compound Distributions. Branching Processes [268] 1. Sums of a Random Number of Variables [268] 2. The Compound Poisson Distribution [270] 3. Infinitely Divisible Distributions [271] 4. Examples for Branching Processes [272] 5. Extinction Probabilities in Branching Processes [274] 6. Problems for Solution [276] XIII Recurrent Events. The Renewal Equation [278] 1. Informal Preparations and Examples [278] 2. Definitions [281] 3. The Basic Relations [285] 4. The Renewal Equation [290] 5. Delayed Recurrent Events [293] 6. The Number of Occurrences of ε [296] *7. Application to the Theory of Success Runs [299] *8. More General Patterns [303] 9. Lack of Memory of Geometric Waiting Times [304] *10. Proof of Theorem 3 of Section 3 [306] 11. Problems for Solution [308] XIV Random Walk and Ruin Problems [311] 1. General Orientation [311] 2. The Classical Ruin Problem [313] 3. Expected Duration of the Game [317] *4. Generating Functions for the Duration of the Game and for the First-Passage Times [318] *5. Explicit Expressions [321] 6. Passage to the Limit; Diffusion Processes [323] *7. Random Walks in the Plane and Space [327] 8. The Generalized One-Dimensional Random Walk (Sequential Sampling) [330] 9. Problems for Solution [334] XV Markov Chains [338] 1. Definition [338] 2. Illustrative Examples [340] 3. Higher Transition Probabilities [347] 4. Closures and Closed Sets [349] 5. Classification of States [351] 6. Ergodic Properties of Irreducible Chains [356] *7. Periodic Chains [360] 8. Transient States [362] 9. Application to Card Shuffling [367] 10. The General Markov Process [368] *11. Miscellany [373] 12. Problems for Solution [376] *XVI Algebraic Treatment of Finite Markov Chains [380] 1. General Theory [380] 2. Examples [384] 3. Random Walk with Reflecting Barriers [388] 4. Transient States; Absorption Probabilities [392] 5. Application to Recurrence Times [395] XVII The Simplest Time-Dependent Stochastic Processes [397] 1. General Orientation [397] 2. The Poisson Process [400] 3. The Pure Birth Process [402] *4. Divergent Birth Processes [404] 5. The Birth and Death Process [407] 6. Exponential Holding Times [411] 7. Waiting Line and Servicing Problems [413] 8. The Backward (Retrospective) Equations [421] 9. Generalization; The Kolmogorov Equations [423] 10. Processes Involving Escapes [428] 11. Problems for Solution [434] Answers to Problems [437] Index [451]
I The Exponential and the Uniform Densities [1] 1. Introduction [1] 2. Densities. Convolutions [3] 3. The Exponential Density [8] 4. Waiting Time Paradoxes. The Poisson Process [11] 5. The Persistence of Bad Luck [15] 6. Waiting Times and Order Statistics [17] 7. The Uniform Distribution [21] 8. Random Splittings [25] 9. Convolutions and Covering Theorems [26] 10. Random Directions [29] 11. The Use of Lebesgue Measure [33] 12. Empirical Distributions [36] 13. Problems for Solution [39] chapter II Special Densities. Randomization [45] 1. Notations and Conventions [45] 2. Gamma Distributions [47] *3. Related Distributions of Statistics [48] 4. Some Common Densities [49] 5. Randomization and Mixtures [53] 6. Discrete Distributions [55] 7. Bessel Functions and Random Walks [58] 8. Distributions on a Circle [61] 9. Problems for Solution [64] CHAPTER III Densities in Higher Dimensions. Normal Densities and Processes [66] 1. Densities [66] 2. Conditional Distributions [71] 3. Return to the Exponential and the Uniform Distributions [74] *4. A Characterization of the Normal Distribution [77] 5. Matrix Notation. The Covariance Matrix [80] 6. Normal Densities and Distributions [83] *7. Stationary Normal Processes [87] 8. Markovian Normal Densities [94] 9. Problems for Solution [99] chapter IV Probability Measures and Spaces [103] 1. Baire Functions [104] 2. Interval Functions and Integrals in Rr [106] 3. σ-Algebras. Measurability [112] 4. Probability Spaces. Random Variables [115] 5. The Extension Theorem [118] 6. Product Spaces. Sequences of Independent Variables [121] 7. Null Sets. Completion [125] chapter V Probability Distributions in Rr [127] 1. Distributions and Expectations [128] 2. Preliminaries [136] 3. Densities [138] 4. Convolutions [143] 5. Symmetrization [148] 6. Integration by Parts. Existence of Moments [150] 7. Chebyshev’s Inequality [151] 8. Further Inequalities. Convex Functions [152] 9. Simple Conditional Distributions. Mixtures [156] *10. Conditional Distributions [160] *11. Conditional Expectations [162] 12. Problems for Solution [165] CHAPTER VI A Survey of some Important Distributions and Processes [169] 1. Stable Distributions in R1 [169] 2. Examples [173] 3. Infinitely Divisible Distributions in R1 [176] 4. Processes with Independent Increments [179] *5. Ruin Problems in Compound Poisson Processes [182] 6. Renewal Processes [184] 7. Examples and Problems [187] 8. Random Walks [190] 9. The Queuing Process [194] 10. Persistent and Transient Random Walks [200] 11. General Markov Chains [205] *12. Martingales [209] 13. Problems for Solution [215] chapter VII Laws of Large Numbers. Applications in Analysis [219] 1. Main Lemma and Notations [219] 2. Bernstein Polynomials. Absolutely Monotone Functions [222] 3. Moment Problems [224] *4. Application to Exchangeable Variables [228] *5. Generalized Taylor Formula and Semi-Groups [230] 6. 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] 10. Problems for Solution [244] CHAPTER VIII The Basic Limit Theorems [247] 1. Convergence of Measures [247] 2. Special Properties [252] 3. Distributions as Operators [254] 4. The Central Limit Theorem [258] *5. Infinite Convolutions [265] 6.Selection Theorems [267] *7. Ergodic Theorems for Markov Chains [270] 8.Regular Variation [275] *9. Asymptotic Properties of Regularly Varying Functions [279] 10.Problems for Solution [284] CHAPTER IX Infinitely Divisible Distributions and Semi-Groups [290] 1. Orientation [290] 2. Convolution Semi-Groups [293] 3. Preparatory Lemmas [296] 4. Finite Variances [298] 5. The Main Theorems [300] 6. Example: Stable Semi-Groups [305] 7. Triangular Arrays with Identical Distributions [308] 8. Domains of Attraction [312] 9. Variable Distributions. The Three-Series Theorem [316] 10. Problems for Solution [318] CHAPTER X Markov Processes and SEmi-Groups [321] 1. The Pseudo-Poisson Type [322] 2. A Variant: Linear Increments [324] 3. Jump Processes [326] 4. Diffusion Processes in R1 [332] 5. The Forward Equation. Boundary Conditions [337] 6. Diffusion in Higher Dimensions [344] 7. Subordinated Processes [345] 8. Markov Processes and Semi-Groups [349] 9. The “Exponential Formula” of Semi-Group Theory [353] 10. Generators. The Backward Equation [356] CHAPTER XI Renewal Theory [358] 1. The Renewal Theorem [358] 2. Proof of the Renewal Theorem [364] *3. Refinements [366] 4. Persistent Renewal Processes [368] 5. The Number Nt of Renewal Epochs [372] 6. Terminating (Transient) Processes [374] 7. Diverse Applications [377] 8. Existence of Limits in Stochastic Processes [379] *9. Renewal Theory on the Whole Line [380] 10. Problems for Solution [385] CHAPTER XII Random Walks in R1 [389] 1. Basic Concepts and Notations [390] 2. Duality. Types of Random Walks [394] 3. Distribution of Ladder Heights. Wiener-Hopf Factorization [398] 3a. The Wiener-Hopf Integral Equation [402] 4. Examples [404] 5. Applications [408] 6. A Combinatorial Lemma [412] 7. Distribution of Ladder Epochs [413] 8. The Arc Sine Laws [417] 9. Miscellaneous Complements [423] 10. Problems for Solution [425] CHAPTER XIII Laplace Transforms. Tauberian Theorems. Resolvents [429] 1. Definitions. The Continuity Theorem [429] 2. Elementary Properties [434] 3. Examples [436] 4. Completely Monotone Functions. Inversion Formulas [439] 5. Tauberian Theorems [442] *6. Stable Distributions [448] *7. Infinitely Divisible Distributions [449] *8. Higher Dimensions [452] 9. Laplace Transforms for Semi-Groups [454] 10. The Hille-Yosida Theorem [458] 11. Problems for Solution [463] CHAPTER XIV Applications of Laplace Transforms [466] 1. The Renewal Equation: Theory [466] 2. Renewal-Type Equations: Examples [468] 3. Limit Theorems Involving Arc Sine Distributions [470] 4. Busy Periods and Related Branching Processes [473] 5. Diffusion Processes [475] 6. Birth-and-Death Processes and Random Walks [479] 7. The Kolmogorov Differential Equations [483] 8. Example: The Pure Birth Process [488] 9. Calculation of Ergodic Limits and of First-Passage Times [491] 10. Problems for Solution [495] CHAPTER XV Characteristic Functions [498] 1. Definition. Basic Properties [498] 2. Special Distributions. Mixtures [502] 2a. Some Unexpected Phenomena [505] 3. Uniqueness. Inversion Formulas [507] 4. Regularity Properties [511] 5. The Central Limit Theorem for Equal Components [515] 6. The Lindeberg Conditions [518] 7. Characteristic Functions in Higher Dimensions [521] *8. Two Characterizations of the Normal Distribution [525] 9. Problems for Solution [526] chapter XVI* Expansions Related to the Central Limit Theorem [531] 1. Notations [532] 2. Expansions for Densities [533] 3. Smoothing [536] 4. Expansions for Distributions [538] 5. The Berry-Esseen Theorems [542] 6. Expansions in the Case of Varying Components [546] 7. Large Deviations [548] chapter XVII Infinitely Divisible Distributions [554] 1. Infinitely Divisible Distributions [554] 2. Canonical Forms. The Main Limit Theorem [58] 2a. Derivatives of Characteristic Functions [565] 3. Examples and Special Properties [566] 4. Special Properties [570] 5. Stable Distributions and Their Domains of Attraction [574] *6. Stable Densities [581] 7. Triangular Arrays [583] *8. The Class L [588] *9. Partial Attraction. “Universal Laws” [590] *10. Infinite Convolutions [592] 11. Higher Dimensions [593] 12. Problems for Solution [595] CHAPTER XVIII Applications of Fourier Methods to Random Walks [598] 1. The Basic Identity [598] *2. Finite Intervals. Wald’s Approximation [601] 3. The Wiener-Hopf Factorization [604] 4. Implications and Applications [609] 5. Two Deeper Theorems [612] 6. Criteria for Persistency [614] 7. Problems for Solution [616] CHAPTER XIX Harmonic Analysis [619] 1. The Parseval Relation [619] 2. Positive Definite Functions [620] 3. Stationary Processes [623] 4. Fourier Series [626] *5. The Poisson Summation Formula [629] 6. Positive Definite Sequences [633] 7. L2 Theory [635] 8. Stochastic Processes and Integrals [641] 9. Problems for Solution [647] Answers to Problems [651] Some Books on Cognate Subjects [655]
| Item type | Home library | Shelving location | Call number | Materials specified | Status | Date due | Barcode | Course reserves |
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Instituto de Matemática, CONICET-UNS | Libros ordenados por tema | 60 F318-2 (Browse shelf) | Vol. 1 | Available | A-295 | ||
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Incluye referencias bibliográficas.
Introduction: The Nature of Probability Theory [1] --
1. The Background [1] --
2. Procedure [3] --
3. “Statistical” Probability [4] --
4. Summary [5] --
5. Historical Note [6] --
I The Sample Space [7] --
1. The Empirical Background [7] --
2. Examples [9] --
3. The Sample Space. Events [13] --
4. Relations among Events [15] --
5. Discrete Sample Spaces [17] --
6. Probabilities in Discrete Sample Spaces: Preparations [19] --
7. The Basic Definitions and Rules [22] --
8. Problems for Solution [24] --
II Elements of Combinatorial Analysis [26] --
1. Preliminaries [26] --
2. Ordered Samples [28] --
3. Examples [30] --
4. Subpopulations and Partitions [32] --
*5. Application to Occupancy Problems [36] --
*5a. Application to Runs [40] --
6. The Hypergeometric Distribution [41] --
7. Examples for Waiting Times [45] --
8. Binomial Coefficients [48] --
9. Stirling’s Formula [50] --
Problems for Solution: --
10. Exercises and Examples [53] --
11. Problems and Complements of a Theoretical Character [57] --
12. Problems and Identities Involving Binomial Coefficients [61] --
*III Fluctuations in Coin Tossing and Random Walks [65] --
1. General Orientation [66] --
2. Problems of Arrangements [69] --
3. Random Walks and Coin Tossing [73] --
4. Reformulation of the Combinatorial Theorems [74] --
5. Probability of Long Leads: The First Arc Sine Law [77] --
6. The Number of Returns to the Origin [81] --
7. An Experimental Illustration [83] --
8. Miscellaneous Complements [85] --
*IV Combination of Events [88] --
1. Union of Events [88] --
2. Application to the Classical Occupancy Problem [91] --
3. The Realization of m among N Events [96] --
4. Application to Matching and Guessing [97] --
5. Miscellany [99] --
6. Problems for Solution [101] --
V Conditional Probability. Stochastic Independence [104] --
1. Conditional Probability [104] --
2. Probabilities Defined by Conditional Probabilities. Urn Models [108] --
3. Stochastic Independence [114] --
4. Repeated Trials [118] --
*5. Applications to Genetics [121] --
*6. Sex-Linked Characters [125] --
*7. Selection [128] --
8. Problems for Solution [129] --
VI The Binomial and the Poisson Distributions [135] --
1. Bernoulli Trials [135] --
2. The Binomial Distribution [136] --
3. The Central Term and the Tails [139] --
4. The Law of Large Numbers [141] --
5. The Poisson Approximation [142] --
6. The Poisson Distribution [146] --
7. Observations Fitting the Poisson Distribution [149] --
8. Waiting Times. The Negative Binomial Distribution [155] --
9. The Multinomial Distribution [157] --
10. Problems for Solution [158] --
VII The Normal Approximation to the Binomial --
Distribution [164] --
1. The Normal Distribution [164] --
2. The DeMoivre-Laplace Limit Theorem [168] --
3. Examples [174] --
4. Relation to the Poisson Approximation [176] --
5. Large Deviations [178] --
6. Problems for Solution [179] --
*VIII Unlimited Sequences of Bernoulli Trials [183] --
1. Infinite Sequences of Trials [183] --
2. Systems of Gambling [185] --
3. The Borel-Cantelli Lemmas [188] --
4. The Strong Law of Large Numbers [189] --
5. The Law of the Iterated Logarithm [191] --
6. Interpretation in Number Theory Language [195] --
7. Problems for Solution [197] --
IX Random Variables; Expectation [199] --
1. Random Variables [199] --
2. Expectations [207] --
3. Examples and Applications [209] --
4. The Variance [213] --
5. Covariance; Variance of a Sum [215] --
6. Chebyshev’s Inequality [219] --
*7. Kolmogorov’s Inequality [220] --
*8. The Correlation Coefficient [221] --
9. Problems for Solution [223] --
X Laws of Large Numbers [228] --
1. Identically Distributed Variables [228] --
*2. Proof of the Law of Large Numbers [231] --
3. The Theory of “Fair” Games [233] --
*4. The Petersburg Game [235] --
5. Variable Distributions [238] --
*6. Applications to Combinatorial Analysis [241] --
*7. The Strong Law of Large Numbers [243] --
8. Problems for Solution [245] --
XI Integral Valued Variables. Generating Functions [248] --
1. Generalities [248] --
2. Convolutions [250] --
3. Application to First Passage and Recurrence Times in Bernoulli Trials [254] --
4. Partial Fraction Expansions [257] --
5. Bivariate Generating Functions [261] --
*6. The Continuity Theorem [262] --
7. Problems for Solution [264] --
*XII Compound Distributions. Branching Processes [268] --
1. Sums of a Random Number of Variables [268] --
2. The Compound Poisson Distribution [270] --
3. Infinitely Divisible Distributions [271] --
4. Examples for Branching Processes [272] --
5. Extinction Probabilities in Branching Processes [274] --
6. Problems for Solution [276] --
XIII Recurrent Events. The Renewal Equation [278] --
1. Informal Preparations and Examples [278] --
2. Definitions [281] --
3. The Basic Relations [285] --
4. The Renewal Equation [290] --
5. Delayed Recurrent Events [293] --
6. The Number of Occurrences of ε [296] --
*7. Application to the Theory of Success Runs [299] --
*8. More General Patterns [303] --
9. Lack of Memory of Geometric Waiting Times [304] --
*10. Proof of Theorem 3 of Section 3 [306] --
11. Problems for Solution [308] --
XIV Random Walk and Ruin Problems [311] --
1. General Orientation [311] --
2. The Classical Ruin Problem [313] --
3. Expected Duration of the Game [317] --
*4. Generating Functions for the Duration of the Game and for the First-Passage Times [318] --
*5. Explicit Expressions [321] --
6. Passage to the Limit; Diffusion Processes [323] --
*7. Random Walks in the Plane and Space [327] --
8. The Generalized One-Dimensional Random Walk --
(Sequential Sampling) [330] --
9. Problems for Solution [334] --
XV Markov Chains [338] --
1. Definition [338] --
2. Illustrative Examples [340] --
3. Higher Transition Probabilities [347] --
4. Closures and Closed Sets [349] --
5. Classification of States [351] --
6. Ergodic Properties of Irreducible Chains [356] --
*7. Periodic Chains [360] --
8. Transient States [362] --
9. Application to Card Shuffling [367] --
10. The General Markov Process [368] --
*11. Miscellany [373] --
12. Problems for Solution [376] --
*XVI Algebraic Treatment of Finite Markov Chains [380] --
1. General Theory [380] --
2. Examples [384] --
3. Random Walk with Reflecting Barriers [388] --
4. Transient States; Absorption Probabilities [392] --
5. Application to Recurrence Times [395] --
XVII The Simplest Time-Dependent Stochastic Processes [397] --
1. General Orientation [397] --
2. The Poisson Process [400] --
3. The Pure Birth Process [402] --
*4. Divergent Birth Processes [404] --
5. The Birth and Death Process [407] --
6. Exponential Holding Times [411] --
7. Waiting Line and Servicing Problems [413] --
8. The Backward (Retrospective) Equations [421] --
9. Generalization; The Kolmogorov Equations [423] --
10. Processes Involving Escapes [428] --
11. Problems for Solution [434] --
Answers to Problems [437] --
Index [451] --
I The Exponential and the Uniform Densities [1] --
1. Introduction [1] --
2. Densities. Convolutions [3] --
3. The Exponential Density [8] --
4. Waiting Time Paradoxes. The Poisson Process [11] --
5. The Persistence of Bad Luck [15] --
6. Waiting Times and Order Statistics [17] --
7. The Uniform Distribution [21] --
8. Random Splittings [25] --
9. Convolutions and Covering Theorems [26] --
10. Random Directions [29] --
11. The Use of Lebesgue Measure [33] --
12. Empirical Distributions [36] --
13. Problems for Solution [39] --
chapter II --
Special Densities. Randomization [45] --
1. Notations and Conventions [45] --
2. Gamma Distributions [47] --
*3. Related Distributions of Statistics [48] --
4. Some Common Densities [49] --
5. Randomization and Mixtures [53] --
6. Discrete Distributions [55] --
7. Bessel Functions and Random Walks [58] --
8. Distributions on a Circle [61] --
9. Problems for Solution [64] --
CHAPTER III --
Densities in Higher Dimensions. Normal Densities and Processes [66] --
1. Densities [66] --
2. Conditional Distributions [71] --
3. Return to the Exponential and the Uniform Distributions [74] --
*4. A Characterization of the Normal Distribution [77] --
5. Matrix Notation. The Covariance Matrix [80] --
6. Normal Densities and Distributions [83] --
*7. Stationary Normal Processes [87] --
8. Markovian Normal Densities [94] --
9. Problems for Solution [99] --
chapter IV --
Probability Measures and Spaces [103] --
1. Baire Functions [104] --
2. Interval Functions and Integrals in Rr [106] --
3. σ-Algebras. Measurability [112] --
4. Probability Spaces. Random Variables [115] --
5. The Extension Theorem [118] --
6. Product Spaces. Sequences of Independent Variables [121] --
7. Null Sets. Completion [125] --
chapter V --
Probability Distributions in Rr [127] --
1. Distributions and Expectations [128] --
2. Preliminaries [136] --
3. Densities [138] --
4. Convolutions [143] --
5. Symmetrization [148] --
6. Integration by Parts. Existence of Moments [150] --
7. Chebyshev’s Inequality [151] --
8. Further Inequalities. Convex Functions [152] --
9. Simple Conditional Distributions. Mixtures [156] --
*10. Conditional Distributions [160] --
*11. Conditional Expectations [162] --
12. Problems for Solution [165] --
CHAPTER VI --
A Survey of some Important Distributions and Processes [169] --
1. Stable Distributions in R1 [169] --
2. Examples [173] --
3. Infinitely Divisible Distributions in R1 [176] --
4. Processes with Independent Increments [179] --
*5. Ruin Problems in Compound Poisson Processes [182] --
6. Renewal Processes [184] --
7. Examples and Problems [187] --
8. Random Walks [190] --
9. The Queuing Process [194] --
10. Persistent and Transient Random Walks [200] --
11. General Markov Chains [205] --
*12. Martingales [209] --
13. Problems for Solution [215] --
chapter VII --
Laws of Large Numbers. Applications in Analysis [219] --
1. Main Lemma and Notations [219] --
2. Bernstein Polynomials. Absolutely Monotone Functions [222] --
3. Moment Problems [224] --
*4. Application to Exchangeable Variables [228] --
*5. Generalized Taylor Formula and Semi-Groups [230] --
6. 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] --
10. Problems for Solution [244] --
CHAPTER VIII --
The Basic Limit Theorems [247] --
1. Convergence of Measures [247] --
2. Special Properties [252] --
3. Distributions as Operators [254] --
4. The Central Limit Theorem [258] --
*5. Infinite Convolutions [265] --
6.Selection Theorems [267] --
*7. Ergodic Theorems for Markov Chains [270] --
8.Regular Variation [275] --
*9. Asymptotic Properties of Regularly Varying Functions [279] --
10.Problems for Solution [284] --
CHAPTER IX --
Infinitely Divisible Distributions and Semi-Groups [290] --
1. Orientation [290] --
2. Convolution Semi-Groups [293] --
3. Preparatory Lemmas [296] --
4. Finite Variances [298] --
5. The Main Theorems [300] --
6. Example: Stable Semi-Groups [305] --
7. Triangular Arrays with Identical Distributions [308] --
8. Domains of Attraction [312] --
9. Variable Distributions. The Three-Series Theorem [316] --
10. Problems for Solution [318] --
CHAPTER X --
Markov Processes and SEmi-Groups [321] --
1. The Pseudo-Poisson Type [322] --
2. A Variant: Linear Increments [324] --
3. Jump Processes [326] --
4. Diffusion Processes in R1 [332] --
5. The Forward Equation. Boundary Conditions [337] --
6. Diffusion in Higher Dimensions [344] --
7. Subordinated Processes [345] --
8. Markov Processes and Semi-Groups [349] --
9. The “Exponential Formula” of Semi-Group Theory [353] --
10. Generators. The Backward Equation [356] --
CHAPTER XI --
Renewal Theory [358] --
1. The Renewal Theorem [358] --
2. Proof of the Renewal Theorem [364] --
*3. Refinements [366] --
4. Persistent Renewal Processes [368] --
5. The Number Nt of Renewal Epochs [372] --
6. Terminating (Transient) Processes [374] --
7. Diverse Applications [377] --
8. Existence of Limits in Stochastic Processes [379] --
*9. Renewal Theory on the Whole Line [380] --
10. Problems for Solution [385] --
CHAPTER XII --
Random Walks in R1 [389] --
1. Basic Concepts and Notations [390] --
2. Duality. Types of Random Walks [394] --
3. Distribution of Ladder Heights. Wiener-Hopf Factorization [398] --
3a. The Wiener-Hopf Integral Equation [402] --
4. Examples [404] --
5. Applications [408] --
6. A Combinatorial Lemma [412] --
7. Distribution of Ladder Epochs [413] --
8. The Arc Sine Laws [417] --
9. Miscellaneous Complements [423] --
10. Problems for Solution [425] --
CHAPTER XIII --
Laplace Transforms. Tauberian Theorems. Resolvents [429] --
1. Definitions. The Continuity Theorem [429] --
2. Elementary Properties [434] --
3. Examples [436] --
4. Completely Monotone Functions. Inversion Formulas [439] --
5. Tauberian Theorems [442] --
*6. Stable Distributions [448] --
*7. Infinitely Divisible Distributions [449] --
*8. Higher Dimensions [452] --
9. Laplace Transforms for Semi-Groups [454] --
10. The Hille-Yosida Theorem [458] --
11. Problems for Solution [463] --
CHAPTER XIV --
Applications of Laplace Transforms [466] --
1. The Renewal Equation: Theory [466] --
2. Renewal-Type Equations: Examples [468] --
3. Limit Theorems Involving Arc Sine Distributions [470] --
4. Busy Periods and Related Branching Processes [473] --
5. Diffusion Processes [475] --
6. Birth-and-Death Processes and Random Walks [479] --
7. The Kolmogorov Differential Equations [483] --
8. Example: The Pure Birth Process [488] --
9. Calculation of Ergodic Limits and of First-Passage Times [491] --
10. Problems for Solution [495] --
CHAPTER XV --
Characteristic Functions [498] --
1. Definition. Basic Properties [498] --
2. Special Distributions. Mixtures [502] --
2a. Some Unexpected Phenomena [505] --
3. Uniqueness. Inversion Formulas [507] --
4. Regularity Properties [511] --
5. The Central Limit Theorem for Equal Components [515] --
6. The Lindeberg Conditions [518] --
7. Characteristic Functions in Higher Dimensions [521] --
*8. Two Characterizations of the Normal Distribution [525] --
9. Problems for Solution [526] --
chapter XVI* --
Expansions Related to the Central Limit Theorem [531] --
1. Notations [532] --
2. Expansions for Densities [533] --
3. Smoothing [536] --
4. Expansions for Distributions [538] --
5. The Berry-Esseen Theorems [542] --
6. Expansions in the Case of Varying Components [546] --
7. Large Deviations [548] --
chapter XVII --
Infinitely Divisible Distributions [554] --
1. Infinitely Divisible Distributions [554] --
2. Canonical Forms. The Main Limit Theorem [58] --
2a. Derivatives of Characteristic Functions [565] --
3. Examples and Special Properties [566] --
4. Special Properties [570] --
5. Stable Distributions and Their Domains of Attraction [574] --
*6. Stable Densities [581] --
7. Triangular Arrays [583] --
*8. The Class L [588] --
*9. Partial Attraction. “Universal Laws” [590] --
*10. Infinite Convolutions [592] --
11. Higher Dimensions [593] --
12. Problems for Solution [595] --
CHAPTER XVIII --
Applications of Fourier Methods to Random Walks [598] --
1. The Basic Identity [598] --
*2. Finite Intervals. Wald’s Approximation [601] --
3. The Wiener-Hopf Factorization [604] --
4. Implications and Applications [609] --
5. Two Deeper Theorems [612] --
6. Criteria for Persistency [614] --
7. Problems for Solution [616] --
CHAPTER XIX --
Harmonic Analysis [619] --
1. The Parseval Relation [619] --
2. Positive Definite Functions [620] --
3. Stationary Processes [623] --
4. Fourier Series [626] --
*5. The Poisson Summation Formula [629] --
6. Positive Definite Sequences [633] --
7. L2 Theory [635] --
8. Stochastic Processes and Integrals [641] --
9. Problems for Solution [647] --
Answers to Problems [651] --
Some Books on Cognate Subjects [655] --
MR, 19,466a (v. 1)
MR, 42 #5292 (v. 2)
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