Proving that Y is a Markov Chain | Finite State Space and Transition Matrix P

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In summary, the conversation discusses a problem involving a Markov Chain and its derived version. The question is whether the derived version is also a Markov Chain and how to prove it formally.
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stochfreak
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Hello, This is my first question here. So, let's see how it goes. I need a proof to a simple problem. The proof seems so obvious that I am finding it hard to prove it. The question is as follows:

Let X = {X_n} (n=0 to infinity) denote a Markov Chain (MC) on a finite state space 'S' and with transition matrix 'P'. Consider the MC, Y = {Y_n} (n=0 to infinity), with the setting, Y_k = X_2k. Prove or disprove that Y is a MC.

On the face of it, it is apparent that Y must be a Markov Chain as it will have to satisfy the Markov property (because X satisfies it and Y is derived from it). But, how can we formally prove this?
 
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Welcome to PF stochfreak.

You can just check the definition of the Markov property (what is it?) using that X satisfies it.
 

What is a Markov Chain?

A Markov chain is a mathematical model that describes a sequence of events or states where the probability of transitioning from one state to another depends only on the current state and not on any previous states.

How do you prove that a system follows the Markov Chain property?

To prove that a system follows the Markov Chain property, we need to show that the probability of transitioning from one state to another is independent of the previous states. This can be done by calculating the conditional probability of transitioning from one state to another given the current state, and showing that it does not depend on any previous states.

What is the importance of proving the Markov Chain property?

Proving the Markov Chain property is important because it allows us to make accurate predictions about the behavior of a system. Knowing that a system follows a Markov Chain allows us to use mathematical tools to analyze and understand its behavior.

What are some common applications of Markov Chains?

Markov Chains have a wide range of applications, including modeling financial markets, predicting weather patterns, analyzing DNA sequences, and simulating complex systems such as traffic flow or population dynamics.

What are the limitations of Markov Chains?

Markov Chains assume that the future state of a system depends only on the current state, which may not always be the case in real-world systems. They also assume that the transition probabilities remain constant over time, which may not hold true in dynamic systems. Additionally, Markov Chains can become computationally complex for large systems with many states and transitions.

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