# Random Walks and Fluctuations (UMAP)

### Author: Geoffrey C. Berresford

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. INTRODUCTION2. A VOTING PROBLEM3. PATHS4. COUNTING PATHS USING THE REFLECTION PRINCIPLE5. RANDOM WALKS6. END-POINT PROBABILITIES7. 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

10. TIME ABOVE ZERO - THE SECOND ARC SINE LAW

The Transformation

**11. A GRAPHICAL EXAMPLE**

12. FURTHER READING

13. SOLUTIONS TO EXERCISES

12. FURTHER READING

13. SOLUTIONS TO EXERCISES

#### Mathematics Topics:

#### Application Areas:

#### Prerequisites:

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