Publication Date
Spring 2021
School
College of Arts and Sciences
Major
Mathematics
Keywords
{Markov chains, stochastic processes, dynamical systems, law of large numbers
Disciplines
Dynamical Systems | Dynamic Systems | Other Statistics and Probability
Recommended Citation
Gentry, Nathanael, "The Fundamental Limit Theorem of Countable Markov Chains" (2021). Senior Honors Theses. 1097.
https://digitalcommons.liberty.edu/honors/1097
Abstract
In 1906, the Russian probabilist A.A. Markov proved that the independence of a sequence of random variables is not a necessary condition for a law of large numbers to exist on that sequence. Markov's sequences -- today known as Markov chains -- touch several deep results in dynamical systems theory and have found wide application in bibliometrics, linguistics, artificial intelligence, and statistical mechanics. After developing the appropriate background, we prove a modern formulation of the law of large numbers (fundamental theorem) for simple countable Markov chains and develop an elementary notion of ergodicity. Then, we apply these chain convergence results to study PageRank and the Google matrix.
Included in
Dynamical Systems Commons, Dynamic Systems Commons, Other Statistics and Probability Commons
