In this post, we will see the book Random Walks ( Mathematical Conversations Part 3) by E. B. Dynkin and V. A. Uspenskii.

About the book

Random Walks is a translation of Part Three of Mathematical Conversations by E. B. Dynkin and V. A. Uspenskii, which was published in the Russian series, Library of the Mathematics Circle. The originality of the exposition and the variety of the problems presented here make this booklet especially useful in stimulating an inventive approach to mathematics. Extensive solutions to all problems are provided.

This booklet deals with some of the more elementary problems in probability theory. The exposition ranges from the simplest examples of a random walk on a line to such more complex examples as random walks through a city, and Markov chains.

The booklet is designed for the reader’s active participation, as the problems are carefully integrated with the text and should be solved in sequence. The reader should have a background of high school algebra.

E. B. DYNKIN, a Professor at Moscow State University, is an eminent mathematician and author, whose specialties are higher algebra, topology, and probability theory. V. A. USPENSKII, a Lecturer at Moscow State University, specializes in mathematical logic.

The book was translated from Russian by Norman D. Whaland, Jr., Olga A. Titelbaum was published in 1963. There is a recent reprint of all three parts combined into one by Dover.

Credits to original uploader.

You can get the book here.

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Contents

Introduction 1

CHAPTER 1. Probability 5

1. Fundamental properties of probability 5
2. Conditional probability 8
3. The formula for complete probability 13

CHAPTER 2. Problems Concerning a Random Walk on an Infinite Line 17

4. Graph of coin tosses 17
5. The triangle of probabilities 18
6. Central elements of the triangle of probabilities 21
7. Estimation of arbitrary elements of the triangle 25
8. The law of the square root of n 26
9. The law of large numbers 34

CHAPTER 3. Random Walks with Finitely Many States 37

10. Random walks on a finite line 37
11. Random walks through a city 39
12. Markov chains 46
13. The meeting problem 47

CHAPTER 4. Random Walks with Infinitely Many States 50

14. Random walks on an infinite path 51
15. The meeting problem 53
16. The infinitely large city with a checkerboard pattern 57

Concluding Remarks 62

Solutions to Problems 64