Persistence of State i in a Markov Chain

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Start date Jun 27, 2011
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In summary, a Markov Chain is a mathematical model that describes the behavior of a system over time, where the future state depends only on the current state. The "persistence of state i" refers to the probability of the system remaining in a particular state for a certain number of time steps. It is calculated by analyzing transition probabilities and is important in understanding the stability and long-term behavior of the system. This concept can be applied in various real-world scenarios, including disease spread, stock market trends, and weather prediction, as well as in machine learning and artificial intelligence algorithms.

Jun 27, 2011

xentity1x

In a Markov chain, show that a state i is persistent if and only if the mean number of visits to the state i is infinite given the chain started in state i.

I thought about looking at the mean recurrence time, but that's all I have so far.

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Jun 27, 2011

lanedance

Homework Helper

start with the definition of persistent

Related to Persistence of State i in a Markov Chain

1. What is a Markov Chain?

A Markov Chain is a mathematical model used to describe the behavior of a system over time. It is a type of stochastic process in which the future state of the system depends only on its current state, and not on any previous states.

2. What is "persistence of state i" in a Markov Chain?

The persistence of state i in a Markov Chain refers to the probability that the system will remain in state i for a certain number of time steps or transitions. It is a measure of the stability or tendency of the system to stay in a particular state.

3. How is "persistence of state i" calculated in a Markov Chain?

The persistence of state i is calculated by analyzing the transition probabilities between states in the Markov Chain. It can be calculated as the product of the transition probabilities from state i to itself over a certain number of time steps.

4. Why is "persistence of state i" important in a Markov Chain?

The persistence of state i is important in understanding the behavior of the system over time. It can provide insights into the stability or long-term behavior of the system, and can also be used to make predictions about the future states of the system.

5. How can "persistence of state i" be applied in real-world scenarios?

The concept of "persistence of state i" in a Markov Chain can be applied in various real-world scenarios, such as studying the spread of diseases, analyzing stock market trends, and predicting weather patterns. It can also be used in machine learning and artificial intelligence algorithms to model and predict the behavior of complex systems.