Random Walks and Fluctuations (UMAP)

This module presents new and simplified proofs of several theorems, including the arc sine law for sojourn times. The theorems are proved by simple geometry transformations of the paths. The graphical presentation has visual appeal and makes the results accessible to probability students at any level. This module will enable students to understand some of the counterintuitive results of probability theory in applications common to everyday life.

Table of Contents:

1. INTRODUCTION

2. A VOTING PROBLEM

3. PATHS

4. COUNTING PATHS USING THE REFLECTION PRINCIPLE

5. RANDOM WALKS

6. END-POINT PROBABILITIES

7. THE PROBABILITY OF STAYING AT LEAST EVEN
7.1 Proof of Theorem 4
The Transformation

8. CHANGES IN LEAD - FEW AND FAR BETWEEN
8.1 Proof of Theorem 5
The Transformation T1
The Transformation T2
The Transformation

9. THE UNEVEN DISTRIBUTION OF RETURNS TO ZERO - THE FIRST ARC SINE LAW

10. TIME ABOVE ZERO - THE SECOND ARC SINE LAW
10.1 Proof of Theorem 7
The Transformation

11. A GRAPHICAL EXAMPLE

12. FURTHER READING

13. SOLUTIONS TO EXERCISES