Markov chain

Definition

A Markov chain is a stochastic process that undergoes transitions between a finite or countable number of states based on certain probabilistic rules. The defining property of a Markov chain is the Markov property, which states that the future state depends only on the present state and not on the sequence of events that preceded it. This concept connects to ergodic theory by analyzing long-term behaviors and stationary processes, as well as how these transformations can be measure-preserving, leading to results in generators and Krieger's theorem.

5 Must Know Facts For Your Next Test

1. Markov chains can be classified into discrete-time and continuous-time processes based on how time is modeled during transitions between states.
2. The transition probabilities in a Markov chain must sum up to 1 for each state, ensuring that the system remains in one of its defined states.
3. Ergodicity in Markov chains guarantees that every state will eventually be reached from any starting state, ensuring long-term predictability.
4. Measure-preserving transformations in Markov chains highlight how the probability distributions remain invariant under certain mappings.
5. Krieger's theorem provides a framework for understanding the relationship between generators and the dynamics of Markov processes, showing how they influence the structure of invariant measures.

Review Questions

• How does the Markov property influence the behavior of a Markov chain over time?
  • The Markov property establishes that the future state of a Markov chain depends solely on its current state, not on how it arrived there. This simplification allows for analysis of long-term behaviors and the development of stationary distributions. As a result, this property enables predictions about future states based only on present conditions, making Markov chains useful in modeling various stochastic processes.
• Discuss the significance of stationary distributions in the context of Markov chains and ergodic theory.
  • Stationary distributions play a crucial role in understanding the long-term behavior of Markov chains. They describe a stable probability distribution that remains unchanged as transitions occur within the chain. In ergodic theory, these distributions are essential for establishing convergence results and ensuring that time averages converge to ensemble averages, thereby providing insight into the overall dynamics of stochastic systems.
• Evaluate how generators relate to Markov chains and their role in Krieger's theorem within ergodic theory.
  • Generators are fundamental to characterizing the infinitesimal behavior of Markov chains by describing how probabilities evolve over infinitesimally small time intervals. In the context of Krieger's theorem, generators help link probabilistic transitions with measure-preserving transformations. This relationship elucidates how stochastic processes behave under different transformations and ensures that invariant measures remain consistent across various systems, ultimately revealing deeper insights into ergodic properties.

Related terms

Transition matrix: A matrix that describes the probabilities of moving from one state to another in a Markov chain.

Ergodicity:

A property of a Markov chain where time averages converge to ensemble averages, indicating that the system explores its state space uniformly over time.

Stationary distribution: A probability distribution that remains unchanged as the Markov chain transitions between states, representing a stable long-term behavior.