## Alexander Toller L&S Sciences

### Computing the Distribution of the Maximal Average of a Random Walk

Suppose that we have a (finite or infinite) series of independent, identically distributed real-valued random variables (increments of time). From this series, we can form a random walk. We can consider the partial sums of this series and analyze the average value of the walk the partial sum divided by the number of increments up to that point at each of its time increments. This project is focused on studying the distribution of the maximum average value of a random walk through a variety of computational algorithms. While there already exists an explicit formula for the probability that the maximum average value is at most any given real number 0 x 1, this formula presents computational challenges for small values of x. What is especially interesting is the asymptotic aspect: how quickly does the probability that the maximal average is at most x approach zero as x approaches zero from the right? Generalizing this to various increment distributions serves as the basis for the project.

#### Message To Sponsor

I am grateful to the SURF program and to those funding this project for their support and encouragement. This program will be an invaluable experience not only in providing an opportunity to broaden my mathematical horizons, but also in honing my own research skills and the communication of my ideas in a scholarly fashion. These would be especially useful in light of my future career goals, which include graduate school in pure math (with particular interest in discrete math, combinatorics, and/or probability) and hopefully teaching/professorship/the tenure track.**Major:**Mathematics

**Mentor:**Jim Pitman

**Sponsor:**Leadership Fund