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Measure theory, probability, and stochastic processes / / Jean-François Le Gall
| Measure theory, probability, and stochastic processes / / Jean-François Le Gall |
| Autore | Le Gall J. F (Jean-François) |
| Pubbl/distr/stampa | Cham, Switzerland : , : Springer, , [2022] |
| Descrizione fisica | 1 online resource (409 pages) |
| Disciplina | 515.42 |
| Collana | Graduate texts in mathematics |
| Soggetto topico |
Measure theory
Probabilities Stochastic processes Teoria de la mesura Probabilitats Processos estocàstics |
| Soggetto genere / forma | Llibres electrònics |
| ISBN |
9783031142055
9783031142048 |
| Formato | Materiale a stampa  |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione | eng |
| Nota di contenuto |
Intro -- Preface -- Contents -- List of Symbols -- Part I Measure Theory -- 1 Measurable Spaces -- 1.1 Measurable Sets -- 1.2 Positive Measures -- 1.3 Measurable Functions -- Operations on Measurable Functions -- 1.4 Monotone Class -- 1.5 Exercises -- 2 Integration of Measurable Functions -- 2.1 Integration of Nonnegative Functions -- 2.2 Integrable Functions -- 2.3 Integrals Depending on a Parameter -- 2.4 Exercises -- 3 Construction of Measures -- 3.1 Outer Measures -- 3.2 Lebesgue Measure -- 3.3 Relation with Riemann Integrals -- 3.4 A Subset of ps: [/EMC pdfmark [/Subtype /Span /ActualText (double struck upper R) /StPNE pdfmark [/StBMC pdfmarkRps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark Which Is Not Measurable -- 3.5 Finite Measures on ps: [/EMC pdfmark [/Subtype /Span /ActualText (double struck upper R) /StPNE pdfmark [/StBMC pdfmarkRps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark and the Stieltjes Integral -- 3.6 The Riesz-Markov-Kakutani Representation Theorem -- 3.7 Exercises -- 4 Lp Spaces -- 4.1 Definitions and the Hölder Inequality -- 4.2 The Banach Space ps: [/EMC pdfmark [/Subtype /Span /ActualText (upper L Superscript p Baseline left parenthesis upper E comma script upper A comma mu right parenthesis) /StPNE pdfmark [/StBMC pdfmarkLp(E,A,μ)ps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark -- 4.3 Density Theorems in Lp Spaces -- 4.4 The Radon-Nikodym Theorem -- 4.5 Exercises -- 5 Product Measures -- 5.1 Product σ-Fields -- 5.2 Product Measures -- 5.3 The Fubini Theorems -- 5.4 Applications -- 5.4.1 Integration by Parts -- 5.4.2 Convolution -- 5.4.3 The Volume of the Unit Ball -- 5.5 Exercises -- 6 Signed Measures -- 6.1 Definition and Total Variation -- 6.2 The Jordan Decomposition -- 6.3 The Duality Between Lp and Lq -- 6.4 The Riesz-Markov-Kakutani Representation Theorem for Signed Measures -- 6.5 Exercises.
7 Change of Variables -- 7.1 The Change of Variables Formula -- 7.2 The Gamma Function -- 7.3 Lebesgue Measure on the Unit Sphere -- 7.4 Exercises -- Part II Probability Theory -- 8 Foundations of Probability Theory -- 8.1 General Definitions -- 8.1.1 Probability Spaces -- 8.1.2 Random Variables -- 8.1.3 Mathematical Expectation -- 8.1.4 An Example: Bertrand's Paradox -- 8.1.5 Classical Laws -- 8.1.6 Distribution Function of a Real Random Variable -- 8.1.7 The σ-Field Generated by a Random Variable -- 8.2 Moments of Random Variables -- 8.2.1 Moments and Variance -- 8.2.2 Linear Regression -- 8.2.3 Characteristic Functions -- 8.2.4 Laplace Transform and Generating Functions -- 8.3 Exercises -- 9 Independence -- 9.1 Independent Events -- 9.2 Independence for σ-Fields and Random Variables -- 9.3 The Borel-Cantelli Lemma -- 9.4 Construction of Independent Sequences -- 9.5 Sums of Independent Random Variables -- 9.6 Convolution Semigroups -- 9.7 The Poisson Process -- 9.8 Exercises -- 10 Convergence of Random Variables -- 10.1 The Different Notions of Convergence -- 10.2 The Strong Law of Large Numbers -- 10.3 Convergence in Distribution -- 10.4 Two Applications -- 10.4.1 The Convergence of Empirical Measures -- 10.4.2 The Central Limit Theorem -- 10.4.3 The Multidimensional Central Limit Theorem -- 10.5 Exercises -- 11 Conditioning -- 11.1 Discrete Conditioning -- 11.2 The Definition of Conditional Expectation -- 11.2.1 Integrable Random Variables -- 11.2.2 Nonnegative Random Variables -- 11.2.3 The Special Case of Square Integrable Variables -- 11.3 Specific Properties of the Conditional Expectation -- 11.4 Evaluation of Conditional Expectation -- 11.4.1 Discrete Conditioning -- 11.4.2 Random Variables with a Density -- 11.4.3 Gaussian Conditioning -- 11.5 Transition Probabilities and Conditional Distributions -- 11.6 Exercises. Part III Stochastic Processes -- 12 Theory of Martingales -- 12.1 Definitions and Examples -- 12.2 Stopping Times -- 12.3 Almost Sure Convergence of Martingales -- 12.4 Convergence in Lp When p> -- 1 -- 12.5 Uniform Integrability and Martingales -- 12.6 Optional Stopping Theorems -- 12.7 Backward Martingales -- 12.8 Exercises -- 13 Markov Chains -- 13.1 Definitions and First Properties -- 13.2 A Few Examples -- 13.2.1 Independent Random Variables -- 13.2.2 Random Walks on ps: [/EMC pdfmark [/Subtype /Span /ActualText (double struck upper Z Superscript d) /StPNE pdfmark [/StBMC pdfmarkZdps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark -- 13.2.3 Simple Random Walk on a Graph -- 13.2.4 Galton-Watson Branching Processes -- 13.3 The Canonical Markov Chain -- 13.4 The Classification of States -- 13.5 Invariant Measures -- 13.6 Ergodic Theorems -- 13.7 Martingales and Markov Chains -- 13.8 Exercises -- 14 Brownian Motion -- 14.1 Brownian Motion as a Limit of Random Walks -- 14.2 The Construction of Brownian Motion -- 14.3 The Wiener Measure -- 14.4 First Properties of Brownian Motion -- 14.5 The Strong Markov Property -- 14.6 Harmonic Functions and the Dirichlet Problem -- 14.7 Harmonic Functions and Brownian Motion -- 14.8 Exercises -- A A Few Facts from Functional Analysis -- Normed Linear Spaces and Banach Spaces -- Hilbert Spaces -- Notes and Suggestions for Further Reading -- References -- Index. |
| Record Nr. | UNINA-9910624396803321 |
Le Gall J. F (Jean-François)
|
||
| Cham, Switzerland : , : Springer, , [2022] | ||
| Materiale a stampa | ||
| Lo trovi qui: Univ. Federico II | ||
| ||
Measure theory, probability, and stochastic processes / / Jean-François Le Gall
| Measure theory, probability, and stochastic processes / / Jean-François Le Gall |
| Autore | Le Gall J. F (Jean-François) |
| Pubbl/distr/stampa | Cham, Switzerland : , : Springer, , [2022] |
| Descrizione fisica | 1 online resource (409 pages) |
| Disciplina | 515.42 |
| Collana | Graduate texts in mathematics |
| Soggetto topico |
Measure theory
Probabilities Stochastic processes Teoria de la mesura Probabilitats Processos estocàstics |
| Soggetto genere / forma | Llibres electrònics |
| ISBN |
9783031142055
9783031142048 |
| Formato | Materiale a stampa |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione | eng |
| Nota di contenuto |
Intro -- Preface -- Contents -- List of Symbols -- Part I Measure Theory -- 1 Measurable Spaces -- 1.1 Measurable Sets -- 1.2 Positive Measures -- 1.3 Measurable Functions -- Operations on Measurable Functions -- 1.4 Monotone Class -- 1.5 Exercises -- 2 Integration of Measurable Functions -- 2.1 Integration of Nonnegative Functions -- 2.2 Integrable Functions -- 2.3 Integrals Depending on a Parameter -- 2.4 Exercises -- 3 Construction of Measures -- 3.1 Outer Measures -- 3.2 Lebesgue Measure -- 3.3 Relation with Riemann Integrals -- 3.4 A Subset of ps: [/EMC pdfmark [/Subtype /Span /ActualText (double struck upper R) /StPNE pdfmark [/StBMC pdfmarkRps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark Which Is Not Measurable -- 3.5 Finite Measures on ps: [/EMC pdfmark [/Subtype /Span /ActualText (double struck upper R) /StPNE pdfmark [/StBMC pdfmarkRps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark and the Stieltjes Integral -- 3.6 The Riesz-Markov-Kakutani Representation Theorem -- 3.7 Exercises -- 4 Lp Spaces -- 4.1 Definitions and the Hölder Inequality -- 4.2 The Banach Space ps: [/EMC pdfmark [/Subtype /Span /ActualText (upper L Superscript p Baseline left parenthesis upper E comma script upper A comma mu right parenthesis) /StPNE pdfmark [/StBMC pdfmarkLp(E,A,μ)ps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark -- 4.3 Density Theorems in Lp Spaces -- 4.4 The Radon-Nikodym Theorem -- 4.5 Exercises -- 5 Product Measures -- 5.1 Product σ-Fields -- 5.2 Product Measures -- 5.3 The Fubini Theorems -- 5.4 Applications -- 5.4.1 Integration by Parts -- 5.4.2 Convolution -- 5.4.3 The Volume of the Unit Ball -- 5.5 Exercises -- 6 Signed Measures -- 6.1 Definition and Total Variation -- 6.2 The Jordan Decomposition -- 6.3 The Duality Between Lp and Lq -- 6.4 The Riesz-Markov-Kakutani Representation Theorem for Signed Measures -- 6.5 Exercises.
7 Change of Variables -- 7.1 The Change of Variables Formula -- 7.2 The Gamma Function -- 7.3 Lebesgue Measure on the Unit Sphere -- 7.4 Exercises -- Part II Probability Theory -- 8 Foundations of Probability Theory -- 8.1 General Definitions -- 8.1.1 Probability Spaces -- 8.1.2 Random Variables -- 8.1.3 Mathematical Expectation -- 8.1.4 An Example: Bertrand's Paradox -- 8.1.5 Classical Laws -- 8.1.6 Distribution Function of a Real Random Variable -- 8.1.7 The σ-Field Generated by a Random Variable -- 8.2 Moments of Random Variables -- 8.2.1 Moments and Variance -- 8.2.2 Linear Regression -- 8.2.3 Characteristic Functions -- 8.2.4 Laplace Transform and Generating Functions -- 8.3 Exercises -- 9 Independence -- 9.1 Independent Events -- 9.2 Independence for σ-Fields and Random Variables -- 9.3 The Borel-Cantelli Lemma -- 9.4 Construction of Independent Sequences -- 9.5 Sums of Independent Random Variables -- 9.6 Convolution Semigroups -- 9.7 The Poisson Process -- 9.8 Exercises -- 10 Convergence of Random Variables -- 10.1 The Different Notions of Convergence -- 10.2 The Strong Law of Large Numbers -- 10.3 Convergence in Distribution -- 10.4 Two Applications -- 10.4.1 The Convergence of Empirical Measures -- 10.4.2 The Central Limit Theorem -- 10.4.3 The Multidimensional Central Limit Theorem -- 10.5 Exercises -- 11 Conditioning -- 11.1 Discrete Conditioning -- 11.2 The Definition of Conditional Expectation -- 11.2.1 Integrable Random Variables -- 11.2.2 Nonnegative Random Variables -- 11.2.3 The Special Case of Square Integrable Variables -- 11.3 Specific Properties of the Conditional Expectation -- 11.4 Evaluation of Conditional Expectation -- 11.4.1 Discrete Conditioning -- 11.4.2 Random Variables with a Density -- 11.4.3 Gaussian Conditioning -- 11.5 Transition Probabilities and Conditional Distributions -- 11.6 Exercises. Part III Stochastic Processes -- 12 Theory of Martingales -- 12.1 Definitions and Examples -- 12.2 Stopping Times -- 12.3 Almost Sure Convergence of Martingales -- 12.4 Convergence in Lp When p> -- 1 -- 12.5 Uniform Integrability and Martingales -- 12.6 Optional Stopping Theorems -- 12.7 Backward Martingales -- 12.8 Exercises -- 13 Markov Chains -- 13.1 Definitions and First Properties -- 13.2 A Few Examples -- 13.2.1 Independent Random Variables -- 13.2.2 Random Walks on ps: [/EMC pdfmark [/Subtype /Span /ActualText (double struck upper Z Superscript d) /StPNE pdfmark [/StBMC pdfmarkZdps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark -- 13.2.3 Simple Random Walk on a Graph -- 13.2.4 Galton-Watson Branching Processes -- 13.3 The Canonical Markov Chain -- 13.4 The Classification of States -- 13.5 Invariant Measures -- 13.6 Ergodic Theorems -- 13.7 Martingales and Markov Chains -- 13.8 Exercises -- 14 Brownian Motion -- 14.1 Brownian Motion as a Limit of Random Walks -- 14.2 The Construction of Brownian Motion -- 14.3 The Wiener Measure -- 14.4 First Properties of Brownian Motion -- 14.5 The Strong Markov Property -- 14.6 Harmonic Functions and the Dirichlet Problem -- 14.7 Harmonic Functions and Brownian Motion -- 14.8 Exercises -- A A Few Facts from Functional Analysis -- Normed Linear Spaces and Banach Spaces -- Hilbert Spaces -- Notes and Suggestions for Further Reading -- References -- Index. |
| Record Nr. | UNISA-996495167103316 |
|
Le Gall J. F (Jean-François)
|
||
| Cham, Switzerland : , : Springer, , [2022] | ||
| Materiale a stampa | ||
| Lo trovi qui: Univ. di Salerno | ||
| ||