Probability Essentials

by Jean Jacod

This introduction to Probability Theory can be used, at the beginning graduate level, for a one-semester course on Probability Theory or for self-direction without benefit of a formal course; the measure theory needed is developed in the text. It will also be useful for students and teachers in related areas such as Finance Theory (Economics), Electrical Engineering, and Operations Research. The text covers the essentials in a directed and lean way with 28 short chapters. Assuming of readers only an undergraduate background in mathematics, it brings them from a starting knowledge of the subject to a knowledge of the basics of Martingale Theory. After learning Probability Theory from this text, the interested student will be ready to continue with the study of more advanced topics, such as Brownian Motion and Ito Calculus, or Statistical Inference. The second edition contains some additions to the text and to the references and some parts are completely rewritten. Contents :- 1. Introduction 2. Axioms of Probability 3. Conditional Probability and Independence 4. Probabilities on a Finite or Countable Space 5. Random Variables on a Countable Space 6. Construction of a Probability Measure 7. Construction of a Probability Measure on R 8. Random Variables 9. Integration with Respect to a Probability Measure 10. Independent Random Variables 11. Probability Distributions on R 12. Probability Distributions on R 13. Characteristic Functions 14. Properties of Characteristic Functions 15. Sums of Independent Random Variables 16. Gaussian Random Variables (The Normal and the Multivariate Normal Distributions) 17. Convergence of Random Variables 18. Weak Convergence 19. Weak Convergence and Characteristic Functions 20. The Laws of Large Numbers 21. The Central Limit Theorem 22. L2 and Hilbert Spaces 23. Conditional Expectation 24. Martingales 25. Supermartingales and Submartingales 26. Martingale Inequalities 27. Martingale Convergence Theorems 28. The Radon-Nikodym Theorem Ref

Genres: Mathematics

254 pages, Paperback

First published November 30, 1999

Reviews

[Evan · Rating 5 out of 5]

Now that's a nice book to take with you to the pool and relax on a hot summer day! All those nice reads!! Highly recommended if you feel guilty with your Ph.D. research, and you want to freshen-up on your measure-theoretic probability (don't we all??)

[Kai · Rating 4 out of 5]

clear exposition

[Alireza Daneshvar · Rating 3 out of 5]

For a review of pure probability basics, this book is a good choice.

[Daniel · Rating 5 out of 5]

Fills a needed niche between extensive electrical-engineering undergrad style probability and more graduate type measure theoretic stochastic processes literature.

[Emanuel · Rating 3 out of 5]

I used this book for two courses at my university. It has the amazing property of actually covering most essentials of probability within a reasonable time frame (even considering the exercises). However, this comes at a very steep cost.

Some parts of the book are outright wrong—so much so that the university had to hand out a 20-page PDF with caveats, additions to the literature, and corrections of the material. It is sad, since it has captured the essence of 'essentials' really well. Its execution was just very lacking. Trying to come up with ideas on how to fix this is probably futile. The last edition was made in 2002, so there likely will not be any corrections made.

None of the following books can do the exact same thing as Probability Essentials, but sometimes the price is just not worth paying.

If you want to review the very basics of probability theory, then I would recommend you to work problems in Blitzstein, J. K., & Hwang, J. (2019). Introduction to Probability. CRC Press, Taylor & Francis Group. Keep in mind this book is quite long, but it has summaries in each chapter and many great exercises to work with.
If you want to review the basics of measure theory, then I would recommend you skim through Schilling, R. L. (2017). Measures, Integrals and Martingales. Cambridge University Press. This book is a monster, but very thorough with solutions to all problems.
If you just want to cover essentials, then I would recommend you to read some of Schilling, R. L. (2021). Measure, Integral, Probability & Process. Technische Universität Dresden. This book is truly dense and captures the essentials of the entirety of probability, but without any exercises.

Edit: There was a corrected printing done in 2004–this is actually the version I'm talking about above. I realise in hindsight, that some of the above is a bit imprecise–I hope that the general message gets through.

[Lokprakash]

Good read Probability Essentials