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AN INTRODUCTION TO RANDOM WALKS

AN INTRODUCTION TO RANDOM WALKSDEREK this paper, we investigate simple RANDOM WALKS inn-dimensionalEuclidean Space. We begin by defining simple RANDOM walk ; we give particularattention to the symmetric RANDOM walk on thed-dimensional integer latticeZd. We proceed to consider returns to the origin, recurrence, the level-crossingphenomenon, and the Gambler s Introduction12. Simple RANDOM walk onZd23. Returns to the Origin24. number of Equalizations onZd85. The Level-Crossing Phenomenon86. Gambler s , a RANDOM walk is a path that is created by some stochastic a simple example, consider a person standing on the integer line who flips a coinand moves one unit to the right if it lands on heads, and one unit to the left if itlands on tails.

2. Simple random walk on Zd 2 3. Returns to the Origin 2 4. Number of Equalizations on Zd 8 5. The Level-Crossing Phenomenon 8 6. Gambler’s Ruin 9 Acknowledgments 11 References 11 1. Introduction Informally, a random walk is a path that is created by some stochastic process. As a simple example, consider a person standing on the integer line ...

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