Markov chains are a fundamental concept in probability theory and have numerous applications in various fields, including engineering, economics, and computer science. In this article, we will provide an in-depth introduction to Markov chains, covering the basic definitions, properties, and applications. We will also discuss the book “Markov Chains” by J.R. Norris, which is a comprehensive resource for anyone looking to learn about Markov chains.
P ( X n + 1 = j ∣ X 0 , X 1 , … , X n ) = P ( X n + 1 = j ∣ X n ) markov chains jr norris pdf
A Markov chain is a mathematical system that undergoes transitions from one state to another according to certain probabilistic rules. The future state of the system depends only on its current state, and not on any of its past states. This property is known as the Markov property. Markov chains are a fundamental concept in probability
The matrix \(P = (p_{ij})\) is called the transition matrix of the Markov chain. Norris, which is a comprehensive resource for anyone
p ij = P ( X n + 1 = j ∣ X n = i )
In other words, the probability of transitioning from state \(i\) to state \(j\) in one step is given by: