Transion probability matrix of independent variables
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Here's a hint.
Assume that at some stage nn, you know that that the maximal value you have observed is kk.
At the next step n+1n+1, what can happen?
Either next ξn+1ξn+1 is smaller than your observed maximum, in which case your chain remains at the same place. (Can you find the probability of ξn+1≤kξn+1≤k?).
Or ξn+1ξn+1 can be higher than your observed maximum. That is, it can be one of {k+1,k+2,k+3,...}{k+1,k+2,k+3,...} and then your chain updates to ξn+1ξn+1 because then this new value is the observed maximum. (With what probabilities will it reach any one of those states?)
Assume that at some stage nn, you know that that the maximal value you have observed is kk.
At the next step n+1n+1, what can happen?
Either next ξn+1ξn+1 is smaller than your observed maximum, in which case your chain remains at the same place. (Can you find the probability of ξn+1≤kξn+1≤k?).
Or ξn+1ξn+1 can be higher than your observed maximum. That is, it can be one of {k+1,k+2,k+3,...}{k+1,k+2,k+3,...} and then your chain updates to ξn+1ξn+1 because then this new value is the observed maximum. (With what probabilities will it reach any one of those states?)
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