This book collects and compares the results — mostly strong theorems which describe the properties of a simple symmetric random walk. The newest problems of limit theorems of probability theory are treated in the very simple case of coin tossing. Using the advantage of this simple situation, the reader can become familiar with limit theorems (especially strong ones) without suffering from technical tools and difficulties. A simple way to the study of the Wiener process is also given, through the study of the random walk. This book presents the most complete study of, and the most elementary way to the study of, the path properties of the Wiener process; and the most elementary way to the study of the strong theorems of probability theory.
Contents:
- Simple Symmetric Random Walk in Z1:
- Introduction of Part I
- Distributions
- Recurrence and the Zero-One Law
- From the Strong Law of Large Numbers to the Law of Iterated Logarithm
- Lévy Classes
- Wiener Process and Invariance Principle
- Increments
- Strassen-Type Theorems
- Distribution of the Local Time
- Local Time and Invariance Principle
- Strong Theorems of the Local Time
- An Embedding Theorem
- Excursions
- A Few Further Results
- Summary of Part I
- Simple Symmetric Random Walk in Zd:
- Recurrence Theorem
- Wiener Process and Invariance Principle
- The Law of Iterated Logarithm
- Local Time
- The Range
- Selfcrossing
- Large Covered Balls
- Speed of Escape
- A Few Further Problems
- Random Walk in Random Environment
- Introduction
- In the First Six Days
- After the Sixth Day
- What Can a Physicist Say About the Local Time ξ(0,n)?
- On the Favourite Value of the RWIRE
- A Few Further Problems
Readership: Graduate students and researchers of probability theory and statistical physics.
