Analysis and Probability

Probability theory is a rapidly expanding field and is used in many areas of science and technology. Beginning from a basis of abstract analysis, this mathematics book develops the knowledge needed for advanced students to develop a complex understanding of probability. The first part of the book systematically presents concepts and results from analysis before embarking on the study of probability theory. The initial section will also be useful for those interested in topology, measure theory, real analysis and functional analysis. The second part of the book presents the concepts, methodology and fundamental results of probability theory. Exercises are included throughout the text, not just at the end, to teach each concept fully as it is explained, including presentations of interesting extensions of the theory. The complete and detailed nature of the book makes it ideal as a reference book or for self-study in probability and related fields.

Covers a wide range of subjects including f-expansions, Fuk-Nagaev inequalities and Markov triples.Provides multiple clearly worked exercises with complete proofs.Guides readers through examples so they can understand and write research papers independently.

Analysis and Probability, 1st Edition

Chapter 1: Elements of Set Theory
1.1 Sets and operations on sets
1.2 Functions and Cartesian products
1.3 Equivalent relations and partial orderings

Chapter 2: Topological Preliminaries
2.1 Construction of some topological spaces
2.2 General properties of topological spaces
2.3 Metric spaces

Chapter 3: Measure Spaces
3.1 Measurable spaces
3.2 Measurable functions
3.3 Denitions and properties of the measure
3.4 Extending certain measures

Chapter 4: The Integral
4.1 Denitions and properties of the integral
4.2 Radon-Nikodým theorem and the Lebesgue decomposition
4.3 The spaces Lp
4.4 Convergence for sequences of measurable functions

Chapter 5: Measures on Product -algebras
5.5 The product of a finite number of measures
5.6 The product of an infnite number of measures

PART TWO: PROBABILITY
Chapter 6: Elementary Notions in Probability Theory
6.1 Events and random variables
6.2 Conditioning and independence

Chapter 7: Distribution Functions and Characteristic Functions
7.1 Distribution functions
7.2 Characteristic functions

Chapter 8: Probabilities on Metric Spaces
8.1 Probabilities in a metric space
8.2 Topology in the space of probabilities

Chapter 9: Central Limit Problem
9.1 Infnitely divisible distribution/characteristic functions
9.2 Convergence to an infnitely divisible distribution/characteristic function

Chapter 10: Sums of Independent Random Variables
10.1 Weak laws of large numbers
10.2 Series of independent random variables
10.3 Strong laws of large numbers
10.4 Laws of the iterated logarithm

Chapter 11: Conditioning
11.1 Conditional expectations, conditional probabilities and conditional independence
11.2 Stopping times and semimartingales

Chapter 12: Ergodicity, Mixing and Stationarity
12.1 Ergodicity and mixing
12.2 Stationary sequences