## Honors Theses

#### Date of Award

2002

#### Document Type

Undergraduate Thesis

#### Department

Mathematics

#### First Advisor

William Staton

#### Relational Format

Dissertation/Thesis

#### Abstract

The general notion of a Markov Chain is introduced in Chapter 1, and a theorem is proven characterizing the two-state Markov Chain. The concept central to the thesis, the Random Walk, is introduced in Chapter 2 and a thorough analysis is presented of a Random Walk in one dimension with a single absoring state. A theorem is proven which provides probabilities of absorption for arbitrary starting points. The theoretical results are then tested by computer simulation, yielding a very satisfying match with the predictions of the theorem. Finally, in Chapter 3 a modification of the Random Walk withou absoring states is presented and analyzed. The expected number of returns to the origin is derived, and again is tested by computer simulation. In this case as well, the simulated tesults provided a nice empirical verification fo the theoretical result. Techniques employed in the work vary acorss the undergraduate curriculum. Linear algebra appears in the power of martices and in the use of eignevalues in Chapter 1. Probability concepts such as independence play a small role. First order linear differential equation techniques are used in chapter 2. Infinite series are evaluated in Chapters 2 and 3, and linear homogeneous recursion equations are solved in Chapter 3. Some use is made of Cataln Numbers and Binomial Coeffecients as well. The computing necessary for the simulations in Chapter 2 and Chapter 3 was done in MATHLAB on the University of Mississippi's supercomputer access through Sweetgum. This thesis was typeset using LATEX.

#### Recommended Citation

Rochester, Joanna Cecile, "A Random Walk, Absorbtion, and Enumeration of Returns" (2002). *Honors Theses*. 2106.

https://egrove.olemiss.edu/hon_thesis/2106

#### Accessibility Status

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