# APTS Applied Stochastic Processes

*(Notes originally produced by Wilfrid Kendall; some material due to Stephen Connor, Christina Goldschmidt, Matt Roberts and Amanda Turner)*

*February 05 2024*

# 1 Introduction

This module will introduce students to two important notions in stochastic processes — reversibility and martingales — identifying the basic ideas, outlining the main results and giving a flavour of some of the important ways in which these notions are used in statistics.

Probability provides one of the major underlying languages of statistics, and purely probabilistic concepts often cross over into the statistical world. So statisticians need to acquire some fluency in the general language of probability.

## 1.1 Learning outcomes

After successfully completing this module an APTS student will be able to:

describe and calculate with the notion of a reversible Markov chain, both in discrete and continuous time;

describe the basic properties of discrete-parameter martingales and check whether the martingale property holds;

recall and apply some significant concepts from martingale theory;

explain how to use Foster-Lyapunov criteria to establish recurrence and speed of convergence to equilibrium for Markov chains.

These outcomes interact interestingly with various topics in applied statistics. However the most important aim of this module is to help students to acquire general awareness of further ideas from probability as and when that might be useful in their further research.

Department of Mathematical Sciences, Durham University, nicholas.georgiou@durham.ac.uk↩︎

Department of Statistical Science, UCL, chak.lo@ucl.ac.uk↩︎