Probability (eBook)

Name: Probability (eBook)

Author(s): Leonardo T. Rolla e Bernardo N. B. de Lima

Pages: 435

Publication: IMPA, 2026

ISBN: 978-85-244-0572-3

Edition: 1

1 Probability spaces
1.1 Some probability models
1.1.1 Sample space
1.1.2 Random events
1.1.3 Probability measure
1.2 Counting and symmetry
1.3 Kolmogorov’s axiomatic formulation
1.4 Measurement spaces
1.4.1 σ-algebras and Borelian sets
1.4.2 Measures
1.5 Exercises

2 Conditional Probability and Independence
2.1 Conditional Probability
2.1.1 Product Rule
2.1.2 Total Probability Law
2.1.3 Bayes Formula
2.2 Independence
2.3 Exercises

3 Random Variables
3.1 Random variables
3.1.1 Induced space and law of a random variable
3.1.2 Distribution function
3.1.3 Quantile function
3.2 Discrete random variables
3.3 Absolutely continuous random variables
3.4 Conditional distribution given an event
3.5 Mixed and singular distributions
3.6 Existence and uniqueness of distributions
3.7 Measurable functions
3.8 Exercises

4 Random vectors
4.1 Random vectors
4.2 Discrete and continuous random vectors
4.3 Sum of independent variables
4.4 Jacobian method
4.5 Sequence of independent variables
4.6 Exercises

5 Mathematical Hope
5.1 Simple random variables
5.2 Mathematical hope
5.3 Approximation and convergence of hope
5.3.1 Monotone Convergence Theorem
5.3.2 Dominated Convergence Theorem
5.4 Conditional hope given an event
5.5 Lebesgue integral
5.5.1 Construction
5.5.2 Main properties
5.5.3 Convergence
5.5.4 Riemann integral and improper integral
5.5.5 Density of measures
5.5.6 Product spaces and iterated integrals
5.6 Exercises

6 Moments and Inequalities
6.1 Moments and variance
6.2 Correlation
6.3 Basic inequalities
6.4 Exercises

7 Convergence of Random Variables
7.1 Modes of convergence
7.2 Borel-Cantelli lemma
7.3 Relationships between modes of convergence
7.4 More on convergence in distributions
7.5 Exercises

8 Law of Large Numbers
8.1 Weak Law of Large Numbers
8.2 Strong Law of Large Numbers
8.3 Strong Kolmogorov Laws
8.4 Some applications
8.5 Exercises

9 Central Limit Theorem
9.1 Central Limit Theorem
9.2 De Moivre-Laplace Theorem
9.3 Lyapunov’s Central Limit Theorem
9.4 Lindeberg’s Central Limit Theorem
9.5 Slutsky’s Theorem and Delta Method
9.6 Exercises

10 Transforms
10.1 Moment generating function
10.2 Characteristic function
10.3 Uniqueness and convergence
10.4 Inversion formula
10.5 Exercises

11 Conditional expectation
11.1 Conditional expectation given a partition
11.2 Conditional probability function
11.3 Conditional density
11.4 Conditional expectation given a σ-algebra
11.5 Radon-Nikodým theorem
11.6 Regular conditional distribution
11.7 Exercises

12 Martingales
12.1 Definitions and examples
12.2 Stop times
12.3 Optional sampling
12.4 Almost certain convergence of martingales
12.5 Convergence of martingales in Lp
12.6 Doob decomposition
12.7 Exercises

13 0-1 Laws and Random Series
13.1 Algebras and spaces of sequences
13.2 Independence of σ-algebras
13.3 Kolmogorov’s 0-1 Law
13.4 Hewitt-Savage’s 0-1 Law
13.5 Convergence of random series
13.6 Exercises

14 Ergodic Theory
14.1 Measure-preserving transformations
14.2 Birkhoff’s Ergodic Theorem
14.3 Ergodic and mixing transformations
14.4 Circle rotation
14.5 von Neumann’s Ergodic Theorem
14.6 Exercises

15 Large Deviations
15.1 Concentration inequality
15.2 Principle of Large Deviations
15.3 Demonstration of Cramér’s Theorem

A Preliminaries
A.1 Calculus
A.2 Taylor Expansion
A.3 Real Analysis

B Stirling formula

C The Real Line and Infinity
C.1 Extended line
C.2 Supreme and upper limit

D Elements of Measure Theory
D.1 Dynkin’s π-λ Theorem
D.2 Carathéodory’s Extension Theorem
D.3 Operations with measurable functions
D.4 Fubini’s and Tonelli’s theorems
D.5 Radon-Nikodým’s theorem
D.6 Regular conditional distribution
D.7 Hölder’s and Minkowski’s inequalities

Bibliography

Contents