Key Concepts of Markov Chains

Markov chains are powerful tools in probability theory, focusing on systems that transition between states based solely on their current position. Understanding their structure, including state spaces and transition probabilities, is essential for analyzing complex processes in engineering and beyond.

1. Definition of Markov Chains

   • A Markov chain is a stochastic process that undergoes transitions between a finite or countable number of states.
   • The future state depends only on the current state and not on the sequence of events that preceded it (Markov property).
   • Markov chains can be classified into discrete-time and continuous-time based on the nature of time progression.

2. State space and transition probabilities

   • The state space is the set of all possible states that the process can occupy.
   • Transition probabilities define the likelihood of moving from one state to another in a single time step.
   • These probabilities must sum to 1 for each state, ensuring that the process will transition to some state.

3. Transition matrix

   • The transition matrix is a square matrix that contains the transition probabilities between states.
   • Each entry in the matrix represents the probability of transitioning from one state to another.
   • The rows of the matrix correspond to the current state, while the columns correspond to the next state.

4. Chapman-Kolmogorov equations

   • These equations relate the transition probabilities over different time intervals.
   • They provide a way to compute the probability of transitioning from one state to another over multiple steps.
   • The equations are fundamental for analyzing the long-term behavior of Markov chains.

5. Classification of states (recurrent, transient, absorbing)

   • Recurrent states are those that, once entered, will be revisited infinitely often.
   • Transient states may be left and not returned to, meaning they can be visited only a finite number of times.
   • Absorbing states are those that, once entered, cannot be left.

6. Irreducibility and periodicity

   • A Markov chain is irreducible if it is possible to reach any state from any other state.
   • Periodicity refers to the property of a state where returns to that state can only occur at multiples of some integer greater than one.
   • Irreducibility and periodicity affect the long-term behavior and classification of states.

7. Stationary distribution

   • A stationary distribution is a probability distribution over states that remains unchanged as the Markov chain evolves.
   • It provides insight into the long-term behavior of the chain, indicating the proportion of time spent in each state.
   • The stationary distribution exists if the Markov chain is irreducible and aperiodic.

8. Limiting distribution

   • The limiting distribution describes the behavior of the Markov chain as time approaches infinity.
   • It represents the probabilities of being in each state after a large number of transitions.
   • The limiting distribution may differ from the stationary distribution in certain cases, particularly in transient states.

9. Ergodic theorem

   • The ergodic theorem states that, for an irreducible and aperiodic Markov chain, the time averages converge to the ensemble averages.
   • This means that long-term averages of state visits will equal the probabilities given by the stationary distribution.
   • It provides a foundation for predicting long-term behavior in Markov chains.

10. Discrete-time vs. continuous-time Markov chains

    • Discrete-time Markov chains update states at fixed time intervals, while continuous-time Markov chains can change states at any point in time.
    • The transition probabilities in discrete-time chains are defined for each time step, whereas continuous-time chains use rates of transition.
    • The analysis methods differ, with continuous-time chains often requiring differential equations.

11. Applications of Markov chains

    • Markov chains are used in various fields, including finance, queueing theory, genetics, and machine learning.
    • They model systems where future states depend only on the current state, such as customer behavior in retail or stock price movements.
    • Markov chains are also foundational in algorithms for optimization and simulation.

12. Markov Chain Monte Carlo (MCMC) methods

    • MCMC methods are a class of algorithms that use Markov chains to sample from probability distributions.
    • They are particularly useful for high-dimensional spaces where direct sampling is challenging.
    • MCMC techniques, such as the Metropolis-Hastings algorithm, allow for approximating complex distributions by generating samples that converge to the desired distribution.