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June 14th, 2016, 03:29 PM  #1 
Member Joined: Jun 2012 Posts: 64 Thanks: 0  Interpreting the limiting distribution of markov chains?
Hi all, I have a few questions on the interpretation and application of the steady state or limiting distribution for a markov chain transition matrix. Let's call this steady state matrix X where X = [0.49 0.51] The states are unemployed and employed from left to right or state 1 and 2. To arrive at this matrix, I used a transition matrix, T, that represented the transition probabilities between the states of unemployment and employment within the year and used the condition that xT = x where x is the stationary distribution. (i.e. x1 = 0.49 and x2=0.51) My first question is to clarify whether X also represents a matrix within a year? From what I understand about the steady state, X, it means that in the long run, the probability of being unemployed is 0.49 within the year? Based on some of the papers I read, we can also use the steady state to represent the "long run proportion of time that the chain spends" in a state. So for the above...does it mean that in the long run, a person will spend half a year unemployed and little more than half a year employed? Also, I have read that I can use X to determine the "expected number of steps(time) until the chain revisits" a state e.g. E[time from state 1 to 1] = 1/x1 . So x1 would be the expected number of steps the chain revisits the unemployment state given the chain started in the unemployment state i.e. 1/0.49 = 2. Does this mean that if I'm unemployed, and leave that state into employment...I expect to be unemployed again after 2 years? Therefore, if apply the above for person's working career before retirement, say for 40 years...it means that the person will spend about 20 years unemployed and expects to be unemployed every 2 years? The paper I'm quoting from is from Karl Sigman. 

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chains, distribution, interpreting, limiting, markov 
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