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# Random Walks and Its Applications

### Bipin Gc ’22, Beloit, Wisconsin

Major: Mathematics

#### Abstract

This is an expository paper on the work of William Feller. William Feller was one of the prominent probabilists of his time. He has contributed immensely in the field of probability theory, and a part of that is covered in this paper.

The main focus will remain in Random Walks. Consider an object moving a unit step either left or right (one-dimensional). Random Walk is the study of the probability of this process. These walks pose intriguing properties. For instance, in case of random walks in one or two dimensions, the walk will always return to the initial position. Using the Central Limit Theorem, we shall show the importance of a one-dimensional random walk on the integer number line. Further, two applications of random walks are shown:

Solving the Classical Ruin Problem. We shall determine the probability of losing for a gambler betting sequentially and the duration of the game.

Comparing two dice board games: snake and ladder, and Ludo. We shall show how two seemingly different games actually fall in the same mathematical category.

Both of the above-mentioned applications describe sequentially one-dimensional random walks. We will further try to draw an abstraction between the two different processes.