Markov Chain Definition

A Markov Chain is a stochastic process that satisfies the Markov property, which states that the future state of the process depends only on the present state and not on the sequence of events that preceded it. In other words, it is a memoryless process where the transition probabilities between states are fixed.

Formally, a Markov Chain is a sequence of random variables $X_0, X_1, X_2, \dots$, where $X_n$ represents the state of the system at step $n$, and it satisfies: $P(X_{n+1} = j \mid X_n = i, X_{n-1} = i_{n-1}, \dots, X_0 = i_0) = P(X_{n+1} = j \mid X_n = i)$ The transition probabilities are usually represented by a transition matrix, where entry $P_{ij}$ denotes the probability of transitioning from state $i$ to state $j$.

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Classification of States

In a Markov Chain, states can be classified into several categories based on their behavior and relationships with other states:

1. Recurrent vs. Transient States:

   • Recurrent State: A state $i$ is recurrent if it will be revisited infinitely often after being entered for the first time. Formally, if $f_{ii} = 1$, where $f_{ii}$ is the probability of eventually returning to $i$.
   • Transient State: A state $i$ is transient if there is a nonzero probability that the process will leave $i$ and never return. Formally, if $f_{ii} < 1$.

2. Periodic vs. Aperiodic States:

   • Periodic State: A state $i$ has a period $d > 1$ if the process can return to $i$ only at multiples of $d$ steps. Formally, the period $d$ is the greatest common divisor (gcd) of all return times $n$ such that $P(X_n = i \mid X_0 = i) > 0$.
   • Aperiodic State: A state $i$ is aperiodic if its period $d = 1$, meaning the process can return to $i$ at irregular intervals.

3. Accessible and Communicating States:

   • Accessible: State $j$ is accessible from state $i$ if there exists $n \geq 0$ such that $P(X_n = j \mid X_0 = i) > 0$. This is denoted as $i \to j$.
   • Communicating: States $i$ and $j$ communicate if $i \to j$ and $j \to i$. A set of states that all communicate forms a communication class.

4. Absorbing State:

   • A state $i$ is absorbing if $P_{ii} = 1$, meaning that once the chain enters state $i$, it cannot leave.

5. Ergodic State:

   • A state is ergodic if it is recurrent, aperiodic, and belongs to a finite Markov chain. Such states ensure the process converges to a unique stationary distribution regardless of the initial state.

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Would you like to see examples or details about Markov chains in specific contexts?

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Five Related Questions

1. What is the role of the transition matrix in defining a Markov Chain?
2. How can we determine if a Markov Chain is irreducible?
3. What are stationary distributions, and how are they related to Markov Chains?
4. Can you explain how Markov Chains are applied in real-world scenarios, like Google PageRank?
5. How do you compute the steady-state probabilities of a Markov Chain?

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Tip

To determine the classification of states in a Markov Chain, construct a state transition diagram and analyze the probabilities and paths between states. This visual approach often simplifies complex systems.