Finding long term Markov behaviors

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In summary: The state-probability row vector after n steps is x_n = x_0 * T^n, which can be written as x_{n-1}*T (a vector times a matrix). If the x_n approach a limiting distribution y, we must have y = y*T, and sum y(j) = 1. This is a simple system of linear equations, called the equilibrium or balance equations.
  • #1
Lolsauce
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Homework Statement



Find the long-term behavior of the Markov system described by the following matrix with initial conditions below: (i.e. find T^infinity)

Find x_infinity

Homework Equations



x_0 = [.3 .3 .4]^TT =

[.5 .4 .1]
[.4 .3 .3]
[.1 .3 .6]

The Attempt at a Solution



So in my attempt to find x_infinity I know that x_n = ( T^n)(x_0)

This means:

[.5^n .4^n .1^n] * [.3 .3 .4]^T
[.4^n .3^n .3^n]
[.1^n .3^n .6^n]

Which I solved to be
[ .3*.5^n .3*.4^n .4*.1^n]
[ .3*.4^n .3*.3^n .4*.3^n]
[ .3*.1^n .3*.3^n .4*.6^n]

But as n approaches infinity all the terms get smaller and all go to zero. Does this seem correct so far?

That would mean:

x_infinity = ( 0 0 0 )^T ? I tried this on my webassign, but it is not correct. Any tips or help in pointing out my mistakes would be great, thanks!
 
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  • #2
Your expression for T^n is wrong. T^n = nth power of T in the sense of matrix multiplication. For example, T^2 = T*T =[[.42,.35,.23],[.35,.35,.31],[.23,.31,.46]], etc. You need to perform vector-matrix multiplications, or else solve the "equilibrium equations" to find the long-run limiting probabilities.

RGV
 
  • #3
Ray Vickson said:
Your expression for T^n is wrong. T^n = nth power of T in the sense of matrix multiplication. For example, T^2 = T*T =[[.42,.35,.23],[.35,.35,.31],[.23,.31,.46]], etc. You need to perform vector-matrix multiplications, or else solve the "equilibrium equations" to find the long-run limiting probabilities.

RGV

Oh I see how I was doing the expression of T^n was wrong. What do you mean by vector-matrix multiplcation or "equilibrium equations"? Do you mean keep finding T^1,T^2,T^3, and see if it approaches a limit?
 
  • #4
The state-probability row vector after n steps is x_n = x_0 * T^n, which can be written as x_{n-1}*T (a vector times a matrix). If the x_n approach a limiting distribution y, we must have y = y*T, and sum y(j) = 1. This is a simple system of linear equations, called the equilibrium or balance equations. I cannot believe that this material is not all in your textbook, but if not, Google "Markov chain" to see various relevant articles.

RGV
 

1. What is "long term Markov behavior"?

Long term Markov behavior refers to a mathematical concept where the future state of a system depends only on its current state, and not on its past history. In other words, the future outcome of a system is independent of the path that has led to it.

2. How do you identify long term Markov behavior in a system?

To identify long term Markov behavior in a system, you would need to analyze the transition probabilities between states. If the probability of transitioning from one state to another remains constant over time, then the system can be considered to exhibit long term Markov behavior.

3. What are the applications of studying long term Markov behavior?

Studying long term Markov behavior can be useful in many fields, including economics, finance, biology, and computer science. It can help predict future outcomes and make decisions based on the current state of a system.

4. Can a system exhibit both short term and long term Markov behavior?

Yes, a system can exhibit both short term and long term Markov behavior. Short term Markov behavior refers to the dependencies between states in the immediate future, while long term Markov behavior looks at the long-term trends and patterns in the system.

5. What are some limitations of using long term Markov behavior in modeling systems?

One limitation is that long term Markov behavior assumes that the system is in a steady state, which may not always be the case. Additionally, it may not account for external factors or sudden changes in the system, which can affect the transition probabilities between states.

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