Website TUEindhoven **TU Eindhoven, Dept. of Mathematics & Computer SCience**

Eindhoven University of Technology (TU/e) is a research university specializing in engineering science & technology.

# BSc Project

## Divergent series in applied probability theory

Many powerful techniques in applied probability theory pertain to the convergence of sequences of random variables. These include e.g. central limit theorems, martingale convergence theorems, large deviation results for estimating tail probabilities, stability results, etc.

The series \(\sum a_n = a_0 + a_1 + \cdots\) is said to be convergent to the sum \(s\) if the partial sums \(s_n = a_1 + \cdots a_n\) converge to a finite number. A series which is not convergent is said to be divergent. The series \(1 – 1 + 1 – 1 + \cdots\), for example, is divergent. With the appropriate definitions, we can assign different values to this divergent series: for example, we can argue that \(1 – 1 + 1 – 1 + \cdots = 1/2, 1/3\), or \(1/4\), all equally well.

On the other hand, the framework of probability theory seems more robust. If we for example define a sequence of random variable \(\{ X_n \}\) where \(X_n = 1\) w.p. \(1/2\) and \(-1\) otherwise, we have \(\mathrm{E}[ \sum X_n ] = 0\). How we precisely define this limit, combined with aperiodicity that the random variable naturally creates, seems to overcome the aforementioned issue of being able to assign different values to the same divergent series.

## Goals

In this BSc project, you will:

- Locate, summarize, and demonstrate all of the most important summation techniques that are available for divergent series.
- Find, summarize, and demonstrate the different definitions of convergence used in probability theory.
- Carry out a solid literature study on where divergent series are used in applied probability.
- Establish whether or not a connection exists between divergent series and probability theory, and discuss the present and potential future uses of divergent series in applied probability theory.