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Probability Theory: A Comprehensive Course 3rd ed. 2020 [Mīkstie vāki]

  • Formāts: Paperback / softback, 716 pages, height x width: 235x155 mm, weight: 1104 g, 24 Illustrations, color; 31 Illustrations, black and white; XIV, 716 p. 55 illus., 24 illus. in color., 1 Paperback / softback
  • Sērija : Universitext
  • Izdošanas datums: 31-Oct-2020
  • Izdevniecība: Springer Nature Switzerland AG
  • ISBN-10: 3030564010
  • ISBN-13: 9783030564018
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Probability Theory: A Comprehensive Course 3rd ed. 2020Palielināt
  • Formāts: Paperback / softback, 716 pages, height x width: 235x155 mm, weight: 1104 g, 24 Illustrations, color; 31 Illustrations, black and white; XIV, 716 p. 55 illus., 24 illus. in color., 1 Paperback / softback
  • Sērija : Universitext
  • Izdošanas datums: 31-Oct-2020
  • Izdevniecība: Springer Nature Switzerland AG
  • ISBN-10: 3030564010
  • ISBN-13: 9783030564018
Citas grāmatas par šo tēmu:
Permanent link: https://www.kriso.lv/db/9783030564018.html
This popular textbook, now in a revised and expanded third edition, presents a comprehensive course in modern probability theory.

Probability plays an increasingly important role not only in mathematics, but also in physics, biology, finance and computer science, helping to understand phenomena such as magnetism, genetic diversity and market volatility, and also to construct efficient algorithms. Starting with the very basics, this textbook covers a wide variety of topics in probability, including many not usually found in introductory books, such as:

  • limit theorems for sums of random variables
  • martingales
  • percolation
  • Markov chains and electrical networks
  • construction of stochastic processes
  • Poisson point process and infinite divisibility
  • large deviation principles and statistical physics
  • Brownian motion
  • stochastic integrals and stochastic differential equations.
The presentation is self-contained and mathematically rigorous, with the material on probability theory interspersed with chapters on measure theory to better illustrate the power of abstract concepts.

This third edition has been carefully extended and includes new features, such as concise summaries at the end of each section and additional questions to encourage self-reflection, as well as updates to the figures and computer simulations. With a wealth of examples and more than 290 exercises, as well as biographical details of key mathematicians, it will be of use to students and researchers in mathematics, statistics, physics, computer science, economics and biology.

1 Basic Measure Theory
1(52)
1.1 Classes of Sets
1(10)
1.2 Set Functions
11(7)
1.3 The Measure Extension Theorem
18(18)
1.4 Measurable Maps
36(9)
1.5 Random Variables
45(8)
2 Independence
53(32)
2.1 Independence of Events
53(8)
2.2 Independent Random Variables
61(8)
2.3 Kolmogorov's 0--1 Law
69(4)
2.4 Example: Percolation
73(12)
3 Generating Functions
85(10)
3.1 Definition and Examples
85(4)
3.2 Poisson Approximation
89(2)
3.3 Branching Processes
91(4)
4 The Integral
95(18)
4.1 Construction and Simple Properties
95(9)
4.2 Monotone Convergence and Fatou's Lemma
104(3)
4.3 Lebesgue Integral Versus Riemann Integral
107(6)
5 Moments and Laws of Large Numbers
113(34)
5.1 Moments
113(8)
5.2 Weak Law of Large Numbers
121(4)
5.3 Strong Law of Large Numbers
125(10)
5.4 Speed of Convergence in the Strong LLN
135(4)
5.5 The Poisson Process
139(8)
6 Convergence Theorems
147(16)
6.1 Almost Sure and Measure Convergence
147(6)
6.2 Uniform Integrability
153(7)
6.3 Exchanging Integral and Differentiation
160(3)
7 Lp-Spaces and the Radon--Nikodym Theorem
163(28)
7.1 Definitions
163(2)
7.2 Inequalities and the Fischer--Riesz Theorem
165(7)
7.3 Hilbert Spaces
172(3)
7.4 Lebesgue's Decomposition Theorem
175(4)
7.5 Supplement: Signed Measures
179(7)
7.6 Supplement: Dual Spaces
186(5)
8 Conditional Expectations
191(22)
8.1 Elementary Conditional Probabilities
191(4)
8.2 Conditional Expectations
195(8)
8.3 Regular Conditional Distribution
203(10)
9 Martingales
213(16)
9.1 Processes, Filtrations, Stopping Times
213(5)
9.2 Martingales
218(5)
9.3 Discrete Stochastic Integral
223(1)
9.4 Discrete Martingale Representation Theorem and the CRR Model
224(5)
10 Optional Sampling Theorems
229(12)
10.1 Doob Decomposition and Square Variation
229(4)
10.2 Optional Sampling and Optional Stopping
233(6)
10.3 Uniform Integrability and Optional Sampling
239(2)
11 Martingale Convergence Theorems and Their Applications
241(16)
11.1 Doob's Inequality
241(2)
11.2 Martingale Convergence Theorems
243(11)
11.3 Example: Branching Process
254(3)
12 Backwards Martingales and Exchangeability
257(16)
12.1 Exchangeable Families of Random Variables
257(6)
12.2 Backwards Martingales
263(3)
12.3 De Finetti's Theorem
266(7)
13 Convergence of Measures
273(30)
13.1 A Topology Primer
274(7)
13.2 Weak and Vague Convergence
281(9)
13.3 Prohorov's Theorem
290(10)
13.4 Application: A Fresh Look at de Finetti's Theorem
300(3)
14 Probability Measures on Product Spaces
303(24)
14.1 Product Spaces
304(3)
14.2 Finite Products and Transition Kernels
307(10)
14.3 Kolmogorov's Extension Theorem
317(5)
14.4 Markov Semigroups
322(5)
15 Characteristic Functions and the Central Limit Theorem
327(40)
15.1 Separating Classes of Functions
327(9)
15.2 Characteristic Functions: Examples
336(8)
15.3 Levy's Continuity Theorem
344(5)
15.4 Characteristic Functions and Moments
349(7)
15.5 The Central Limit Theorem
356(9)
15.6 Multidimensional Central Limit Theorem
365(2)
16 Infinitely Divisible Distributions
367(24)
16.1 Levy--Khinchin Formula
367(14)
16.2 Stable Distributions
381(10)
17 Markov Chains
391(44)
17.1 Definitions and Construction
391(8)
17.2 Discrete Markov Chains: Examples
399(5)
17.3 Discrete Markov Processes in Continuous Time
404(7)
17.4 Discrete Markov Chains: Recurrence and Transience
411(4)
17.5 Application: Recurrence and Transience of Random Walks
415(8)
17.6 Invariant Distributions
423(6)
17.7 Stochastic Ordering and Coupling
429(6)
18 Convergence of Markov Chains
435(26)
18.1 Periodicity of Markov Chains
435(4)
18.2 Coupling and Convergence Theorem
439(6)
18.3 Markov Chain Monte Carlo Method
445(8)
18.4 Speed of Convergence
453(8)
19 Markov Chains and Electrical Networks
461(32)
19.1 Harmonic Functions
462(3)
19.2 Reversible Markov Chains
465(2)
19.3 Finite Electrical Networks
467(6)
19.4 Recurrence and Transience
473(7)
19.5 Network Reduction
480(8)
19.6 Random Walk in a Random Environment
488(5)
20 Ergodic Theory
493(22)
20.1 Definitions
493(4)
20.2 Ergodic Theorems
497(3)
20.3 Examples
500(2)
20.4 Application: Recurrence of Random Walks
502(4)
20.5 Mixing
506(4)
20.6 Entropy
510(5)
21 Brownian Motion
515(58)
21.1 Continuous Versions
515(7)
21.2 Construction and Path Properties
522(7)
21.3 Strong Markov Property
529(3)
21.4 Supplement: Feller Processes
532(3)
21.5 Construction via L2-Approximation
535(9)
21.6 The Space C([ 0, ∞))
544(2)
21.7 Convergence of Probability Measures on C([ 0, ∞))
546(3)
21.8 Donsker's Theorem
549(4)
21.9 Pathwise Convergence of Branching Processes
553(7)
21.10 Square Variation and Local Martingales
560(13)
22 Law of the Iterated Logarithm
573(14)
22.1 Iterated Logarithm for the Brownian Motion
573(3)
22.2 Skorohod's Embedding Theorem
576(7)
22.3 Hartman--Wintner Theorem
583(4)
23 Large Deviations
587(24)
23.1 Cramer's Theorem
588(6)
23.2 Large Deviations Principle
594(4)
23.3 Sanov's Theorem
598(5)
23.4 Varadhan's Lemma and Free Energy
603(8)
24 The Poisson Point Process
611(24)
24.1 Random Measures
611(5)
24.2 Properties of the Poisson Point Process
616(11)
24.3 The Poisson-Dirichlet Distribution
627(8)
25 The Ito Integral
635(30)
25.1 Ito Integral with Respect to Brownian Motion
635(9)
25.2 Ito Integral with Respect to Diffusions
644(4)
25.3 The Ito Formula
648(9)
25.4 Dirichlet Problem and Brownian Motion
657(2)
25.5 Recurrence and Transience of Brownian Motion
659(6)
26 Stochastic Differential Equations
665(26)
26.1 Strong Solutions
665(10)
26.2 Weak Solutions and the Martingale Problem
675(7)
26.3 Weak Uniqueness via Duality
682(9)
References 691(8)
Notation Index 699(4)
Name Index 703(4)
Subject Index 707
Achim Klenke is a professor at the Johannes Gutenberg University in Mainz, Germany. He is known for his work on interacting particle systems, stochastic analysis, and branching processes, in particular for his pioneering work with Leonid Mytnik on infinite rate mutually catalytic branching processes.