Markov chains

Definition

Markov chains are mathematical systems that undergo transitions from one state to another within a finite or countable number of possible states, where the probability of each transition depends only on the current state and not on the previous states. This property, known as the Markov property, makes them particularly useful in modeling random processes, including random walks. In the context of random walks, Markov chains can help understand the behavior of particles as they move randomly on a graph or in space.

5 Must Know Facts For Your Next Test

1. Markov chains can be either discrete or continuous based on the nature of their state space and transition probabilities.
2. In a Markov chain, the future state is independent of past states given the present state, which simplifies analysis and computation.
3. Random walks are often modeled as Markov chains, where each step represents a transition between states based on certain probabilities.
4. The long-term behavior of a Markov chain can be studied using its stationary distribution, which describes the probabilities of being in each state after many transitions.
5. Applications of Markov chains extend beyond random walks and include areas like finance, physics, and computer science for modeling various stochastic processes.

Review Questions

• How does the Markov property influence the behavior of random walks?
  • The Markov property asserts that the future behavior of a random walk depends only on its current position, not on how it arrived there. This means that at each step, the random walker makes decisions based solely on their current state. As a result, random walks can be effectively modeled using Markov chains, allowing for simplified analysis and prediction of long-term behaviors based on transition probabilities.
• Compare and contrast the concepts of transition matrices and state spaces in Markov chains.
  • Transition matrices and state spaces are fundamental components of Markov chains. The state space is the complete set of possible states that a system can be in, while the transition matrix captures the probabilities of moving between those states. Understanding both concepts is crucial since they work together to describe how a Markov chain operates: the state space identifies what states are possible, and the transition matrix determines how likely it is to move from one state to another during transitions.
• Evaluate how ergodicity in Markov chains can impact predictions made from random walk models.
  • Ergodicity ensures that a Markov chain will converge to a stationary distribution over time, regardless of the starting state. This property is essential for making reliable predictions from random walk models because it implies that long-term outcomes can be understood irrespective of initial conditions. By leveraging ergodicity, analysts can focus on determining stationary distributions to inform decisions or strategies based on expected long-term behaviors rather than short-term fluctuations.

Related terms

State Space: The set of all possible states that a Markov chain can occupy.

Transition Matrix: A matrix that represents the probabilities of transitioning from one state to another in a Markov chain.

Ergodicity: A property of a Markov chain where, given enough time, it will converge to a unique stationary distribution regardless of the initial state.