An introduction to probability theory and its applications / William Feller.
Series Wiley mathematical statistics series ; Wiley series in probability and mathematical statisticsEditor: New York : Wiley, c1950-1966Descripción: 2 v. ; 24 cmOtra clasificación: 60-01I 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 [10] 5. The Persistence of Bad Luck [15] 6. Waiting Times and Order Statistics [17] 7. The Uniform Distribution [20] 8. Random Splittings [24] 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] II Special Densities. Randomization [44] 1. Notations and Conventions [44] 2. Gamma Distributions [46] *3. Related Distributions of Statistics [47] 4. Some Common Densities [48] 5. Randomization and Mixtures [52] 6. Discrete Distributions [55] 7. Bessel Functions and Random Walks [57] 8. Distributions on a Circle [60] 9. Problems for Solution [63] III Densities in Higher Dimensions. Normal Densities and Processes [65] 1. Densities [65] 2. Conditional Distributions [70] 3. Return to the Exponential and the Uniform Distributions [73] *4. A Characterization of the Normal Distribution [77] 5. Matrix Notation. The Covariance Matrix [80] 6. Normal Densities and Distributions [82] 6a. Appendix: Rotations [86] *7. Stationary Normal Processes [87] 8. Markovian Normal Densities [93] 9. Problems for Solution [98] IV Probability Measures and Spaces [101] 1. Baire Functions [102] 2. Interval Functions and Integrals in Rr [104] 3. Probability Measures and Spaces [110] 4. Random Variables. Expectations [112] 5. The Extension Theorem [116] 6. Product Spaces. Sequences of Independent Variables [118] 7. Null Sets. Completion [123] V Probability Distributions in Rr [125] 1. Distributions and Expectations [126] 2. Preliminaries [133] 3. Densities [136] *3a. Singular Distributions [138] 4. Convolutions [140] 5. Symmetrization [146] 6. Integration by Parts. Existence of Moments [148] 7. Chebyshev’s Inequality [149] 8. Further Inequalities. Convex Functions [150] 9. Simple Conditional Distributions. Mixtures [154] *10. Conditional Distributions [157] *10a. Conditional Expectations [160] 11. Problems for Solution [162] VI A Survey of Some Important Distributions and Processes [165] 1. Stable Distributions in R1 [165] 2. Examples [170] 3. Infinitely Divisible Distributions in R1 [173] 4. Processes with Independent Increments [177] *5. Ruin Problems in Compound Poisson Processes [179] 6. Renewal Processes [181] 7. Examples and Problems [185] 8. Random Walks [189] 9. The Queuing Process [193] 10. Persistent and Transient Random Walks [199] 11. General Markov Chains [205] *12. Martingales [210] 13. Problems for Solution [215] VII Laws of Large Numbers. Applications in Analysis [218] 1. Main Lemma and Notations [218] 2. Bernstein Polynomials. Absolutely Monotone Functions [220] 3. Moment Problems [222] *4. Application to Exchangeable Variables [225] *5. Generalized Taylor Formula and Semi-groups [227] 6. Inversion Formulas for Laplace Transforms [229] *7. Laws of Large Numbers for Identically Distributed Variables [231] 8. Strong Laws for Martingales [234] 9. Problems for Solution [239] VIII The Basic Limit Theorems [241] 1. Convergence of Measures [241] 2. Special Properties [245] 3. Distributions as Operators [248] 4. The Central Limit Theorem [252] *5. Infinite Convolutions [259] 6. Selection Theorems [260] *7. Ergodic Theorems for Markov Chains [264] 8. Regular Variation [268] *9. Asymptotic Properties of Regularly Varying Functions [272] 10. Problems for Solution [276] IX Infinitely Divisible Distributions and Semi-groups [281] 1. Orientation [281] 2. Convolution Semi-groups [284] 3. Preparatory Lemmas [287] 4. Finite Variances [289] 5. The Main Theorems [291] *5a. Discontinuous Semi-groups [296] 6. Example: Stable Semi-groups [296] 7. Triangular Arrays [298] 8. Domains of Attraction [302] 9. Variable Distributions. The Three-series Theorem [306] 10. Problems for Solution [309] X Markov Processes and Semi-groups [311] 1. The Pseudo-Poisson Type [312] 2. A Variant: Linear Increments [314] 3. Jump Processes [316] 4. Diffusion Processes in R1 [320] 5. The Forward Equation: Boundary Conditions [326] 6. Diffusion in Higher Dimensions [331] 7. Subordinated Processes [333] 8. Markov Processes and Semi-groups [337] 9. The “Exponential Formula” of Semi-group Theory [341] 10. Generators. The Backward Equation [343] XI Renewal Theory [346] 1. The Renewal Theorem [346] *2. The Equation ζ = F ★ ζ [351] 3. Persistent Renewal Processes [353] 4. Refinements [357] 5. The Central Limit Theorem [358] 6. Terminating (Transient) Processes [360] 7. Applications [363] 8. Existence of Limits in Stochastic Processes [365] *9. Renewal Theory on the Whole Line [367] 10. Problems for Solution [371] XII Random Walks in R1 [373] 1. Notations and Conventions [374] 2. Duality [377] 3. Distribution of Ladder Heights. Wiener-Hopf Factorization [381] 3a. The Wiener-Hopf Integral Equation [385] 4. Examples [386] 5. Applications [390] 6. A Combinatorial Lemma [393] 7. Distribution of Ladder Epochs [394] 8. The Arc Sine Laws [397] 9. Miscellaneous Complements [402] 10. Problems for Solution [403] XIII Laplace Transforms. Tauberian Theorems. Resolvents [407] 1. Definitions. The Continuity Theorem [407] 2. Elementary Properties [411] 3. Examples [413] 4. Completely Monotone Functions. Inversion Formulas [415] 5. Tauberian Theorems [418] *6. Stable Distributions [424] *7. Infinitely Divisible Distributions [425] *8. Higher Dimensions [428] 9. Laplace Transforms for Semi-groups [429] 10. The Hille-Yosida Theorem [433] 11. Problems for Solution [437] XIV Applications of Laplace Transforms [441] 1. The Renewal Equation: Theory [441] 2. Renewal-Type Equations: Examples [443] 3. Limit Theorems Involving Arc Sine Distributions [445] 4. Busy Periods and Related Branching Processes [448] 5. Diffusion Processes [450] 6. Birth-and-Death Processes and Random Walks [454] 7. The Kolmogorov Differential Equations [457] 8. Example: The Pure Birth Process [463] 9. Calculation of P(oo) and of First-passage Times [465] 10. Problems for Solution [469] XV Characteristic Functions [472] 1. Definition. Basic Properties [472] 2. Special Densities. Mixtures [475] 3. Uniqueness. Inversion Formulas [480] 4. Regularity Properties [484] 5. The Central Limit Theorem for Equal Components [487] 6. The Lindeberg Conditions [491] 7. Characteristic Functions in Higher Dimensions [494] *8. Two Characterizations of the Normal Distribution [498] 9. Problems for Solution [500] XVI Expansions Related to the Central Limit Theorem [504] 1. Notations [505] 2. Expansions for Densities [506] 3. Smoothing [510] 4. Expansions for Distributions [512] 5. The Berry-Esséen Theorem [515] 6. Large Deviations [517] 7. Unequal Components [521] 8. Problems for Solution [524] XVII Infinitely Divisible Distributions [526] 1. A Convergence Theorem [526] 2. Infinitely Divisible Distributions [531] 3. Examples and Special Properties [536] 4. Stable Characteristic Functions [540] 5. Domains of Attraction [543] *6. Stable Densities [548] 7. Triangular Arrays [550] *8. The Class L [553] *9. Partial Attraction. “Universal Laws” [555] *10. Infinite Convolutions [558] 11. Higher Dimensions [559] 12. Problems for Solution [560] XVIII Applications of Fourier Methods to Random Walks [564] 1. The Basic Identity [564] *2. Finite Intervals. Wald’s Approximation [566] 3. Wiener-Hopf Factorization [569] 4. Discussion and Applications [572] *5. Refinements [575] 6. Returns to the Origin [576] 7. Criteria for Persistency [577] 8. Problems for Solution [580] XIX Harmonic Analysis [582] 1. The Parseval Relation [582] 2. Positive Definite Functions [584] 3. Stationary Processes [586] 4. Fourier Series [589] *5. The Poisson Summation Formula [592] 6. Positive Definite Sequences [595] 7. L2 Theory [597] 8. Stochastic Processes and Integrals [602] 9. Problems for Solution [608] Answers to Problems [611] Some Books on Cognate Subjects [615] Index [619]
| Item type | Home library | Shelving location | Call number | Materials specified | Copy number | Status | Date due | Barcode | Course reserves |
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Libros
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Instituto de Matemática, CONICET-UNS | Libros ordenados por tema | 60 F318 (Browse shelf) | v. 2 | Available | A-2783 | |||
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Libros
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Instituto de Matemática, CONICET-UNS | Libros ordenados por tema | 60 F318 (Browse shelf) | v. 2 | Ej. 2 | Available | A-2784 |
La biblioteca sólo posee el v. 2. AR-BbIMB
Notas bibliográficas al pie. Bibliografía: v. 2, p. 615-616.
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 [10] --
5. The Persistence of Bad Luck [15] --
6. Waiting Times and Order Statistics [17] --
7. The Uniform Distribution [20] --
8. Random Splittings [24] --
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] --
II Special Densities. Randomization [44] --
1. Notations and Conventions [44] --
2. Gamma Distributions [46] --
*3. Related Distributions of Statistics [47] --
4. Some Common Densities [48] --
5. Randomization and Mixtures [52] --
6. Discrete Distributions [55] --
7. Bessel Functions and Random Walks [57] --
8. Distributions on a Circle [60] --
9. Problems for Solution [63] --
III Densities in Higher Dimensions. Normal Densities and Processes [65] --
1. Densities [65] --
2. Conditional Distributions [70] --
3. Return to the Exponential and the Uniform Distributions [73] --
*4. A Characterization of the Normal Distribution [77] --
5. Matrix Notation. The Covariance Matrix [80] --
6. Normal Densities and Distributions [82] --
6a. Appendix: Rotations [86] --
*7. Stationary Normal Processes [87] --
8. Markovian Normal Densities [93] --
9. Problems for Solution [98] --
IV Probability Measures and Spaces [101] --
1. Baire Functions [102] --
2. Interval Functions and Integrals in Rr [104] --
3. Probability Measures and Spaces [110] --
4. Random Variables. Expectations [112] --
5. The Extension Theorem [116] --
6. Product Spaces. Sequences of Independent Variables [118] --
7. Null Sets. Completion [123] --
V Probability Distributions in Rr [125] --
1. Distributions and Expectations [126] --
2. Preliminaries [133] --
3. Densities [136] --
*3a. Singular Distributions [138] --
4. Convolutions [140] --
5. Symmetrization [146] --
6. Integration by Parts. Existence of Moments [148] --
7. Chebyshev’s Inequality [149] --
8. Further Inequalities. Convex Functions [150] --
9. Simple Conditional Distributions. Mixtures [154] --
*10. Conditional Distributions [157] --
*10a. Conditional Expectations [160] --
11. Problems for Solution [162] --
VI A Survey of Some Important Distributions and Processes [165] --
1. Stable Distributions in R1 [165] --
2. Examples [170] --
3. Infinitely Divisible Distributions in R1 [173] --
4. Processes with Independent Increments [177] --
*5. Ruin Problems in Compound Poisson Processes [179] --
6. Renewal Processes [181] --
7. Examples and Problems [185] --
8. Random Walks [189] --
9. The Queuing Process [193] --
10. Persistent and Transient Random Walks [199] --
11. General Markov Chains [205] --
*12. Martingales [210] --
13. Problems for Solution [215] --
VII Laws of Large Numbers. Applications in Analysis [218] --
1. Main Lemma and Notations [218] --
2. Bernstein Polynomials. Absolutely Monotone --
Functions [220] --
3. Moment Problems [222] --
*4. Application to Exchangeable Variables [225] --
*5. Generalized Taylor Formula and Semi-groups [227] --
6. Inversion Formulas for Laplace Transforms [229] --
*7. Laws of Large Numbers for Identically Distributed Variables [231] --
8. Strong Laws for Martingales [234] --
9. Problems for Solution [239] --
VIII The Basic Limit Theorems [241] --
1. Convergence of Measures [241] --
2. Special Properties [245] --
3. Distributions as Operators [248] --
4. The Central Limit Theorem [252] --
*5. Infinite Convolutions [259] --
6. Selection Theorems [260] --
*7. Ergodic Theorems for Markov Chains [264] --
8. Regular Variation [268] --
*9. Asymptotic Properties of Regularly Varying Functions [272] --
10. Problems for Solution [276] --
IX Infinitely Divisible Distributions and Semi-groups [281] --
1. Orientation [281] --
2. Convolution Semi-groups [284] --
3. Preparatory Lemmas [287] --
4. Finite Variances [289] --
5. The Main Theorems [291] --
*5a. Discontinuous Semi-groups [296] --
6. Example: Stable Semi-groups [296] --
7. Triangular Arrays [298] --
8. Domains of Attraction [302] --
9. Variable Distributions. The Three-series Theorem [306] --
10. Problems for Solution [309] --
X Markov Processes and Semi-groups [311] --
1. The Pseudo-Poisson Type [312] --
2. A Variant: Linear Increments [314] --
3. Jump Processes [316] --
4. Diffusion Processes in R1 [320] --
5. The Forward Equation: Boundary Conditions [326] --
6. Diffusion in Higher Dimensions [331] --
7. Subordinated Processes [333] --
8. Markov Processes and Semi-groups [337] --
9. The “Exponential Formula” of Semi-group Theory [341] --
10. Generators. The Backward Equation [343] --
XI Renewal Theory [346] --
1. The Renewal Theorem [346] --
*2. The Equation ζ = F ★ ζ [351] --
3. Persistent Renewal Processes [353] --
4. Refinements [357] --
5. The Central Limit Theorem [358] --
6. Terminating (Transient) Processes [360] --
7. Applications [363] --
8. Existence of Limits in Stochastic Processes [365] --
*9. Renewal Theory on the Whole Line [367] --
10. Problems for Solution [371] --
XII Random Walks in R1 [373] --
1. Notations and Conventions [374] --
2. Duality [377] --
3. Distribution of Ladder Heights. Wiener-Hopf Factorization [381] --
3a. The Wiener-Hopf Integral Equation [385] --
4. Examples [386] --
5. Applications [390] --
6. A Combinatorial Lemma [393] --
7. Distribution of Ladder Epochs [394] --
8. The Arc Sine Laws [397] --
9. Miscellaneous Complements [402] --
10. Problems for Solution [403] --
XIII Laplace Transforms. Tauberian Theorems. Resolvents [407] --
1. Definitions. The Continuity Theorem [407] --
2. Elementary Properties [411] --
3. Examples [413] --
4. Completely Monotone Functions. Inversion Formulas [415] --
5. Tauberian Theorems [418] --
*6. Stable Distributions [424] --
*7. Infinitely Divisible Distributions [425] --
*8. Higher Dimensions [428] --
9. Laplace Transforms for Semi-groups [429] --
10. The Hille-Yosida Theorem [433] --
11. Problems for Solution [437] --
XIV Applications of Laplace Transforms [441] --
1. The Renewal Equation: Theory [441] --
2. Renewal-Type Equations: Examples [443] --
3. Limit Theorems Involving Arc Sine Distributions [445] --
4. Busy Periods and Related Branching Processes [448] --
5. Diffusion Processes [450] --
6. Birth-and-Death Processes and Random Walks [454] --
7. The Kolmogorov Differential Equations [457] --
8. Example: The Pure Birth Process [463] --
9. Calculation of P(oo) and of First-passage Times [465] --
10. Problems for Solution [469] --
XV Characteristic Functions [472] --
1. Definition. Basic Properties [472] --
2. Special Densities. Mixtures [475] --
3. Uniqueness. Inversion Formulas [480] --
4. Regularity Properties [484] --
5. The Central Limit Theorem for Equal Components [487] --
6. The Lindeberg Conditions [491] --
7. Characteristic Functions in Higher Dimensions [494] --
*8. Two Characterizations of the Normal Distribution [498] --
9. Problems for Solution [500] --
XVI Expansions Related to the Central Limit Theorem [504] --
1. Notations [505] --
2. Expansions for Densities [506] --
3. Smoothing [510] --
4. Expansions for Distributions [512] --
5. The Berry-Esséen Theorem [515] --
6. Large Deviations [517] --
7. Unequal Components [521] --
8. Problems for Solution [524] --
XVII Infinitely Divisible Distributions [526] --
1. A Convergence Theorem [526] --
2. Infinitely Divisible Distributions [531] --
3. Examples and Special Properties [536] --
4. Stable Characteristic Functions [540] --
5. Domains of Attraction [543] --
*6. Stable Densities [548] --
7. Triangular Arrays [550] --
*8. The Class L [553] --
*9. Partial Attraction. “Universal Laws” [555] --
*10. Infinite Convolutions [558] --
11. Higher Dimensions [559] --
12. Problems for Solution [560] --
XVIII Applications of Fourier Methods to Random Walks [564] --
1. The Basic Identity [564] --
*2. Finite Intervals. Wald’s Approximation [566] --
3. Wiener-Hopf Factorization [569] --
4. Discussion and Applications [572] --
*5. Refinements [575] --
6. Returns to the Origin [576] --
7. Criteria for Persistency [577] --
8. Problems for Solution [580] --
XIX Harmonic Analysis [582] --
1. The Parseval Relation [582] --
2. Positive Definite Functions [584] --
3. Stationary Processes [586] --
4. Fourier Series [589] --
*5. The Poisson Summation Formula [592] --
6. Positive Definite Sequences [595] --
7. L2 Theory [597] --
8. Stochastic Processes and Integrals [602] --
9. Problems for Solution [608] --
Answers to Problems [611] --
Some Books on Cognate Subjects [615] --
Index [619] --
MR, 12,424a (v. 1)
MR, 35 #1048 (v. 2)
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