The entrance probability theorem shows that if we start from a position say i then the probability of arriving at another position say j for the first time within a time period n is fij(n)
The equation thus becomes:
= P [ Xn = j, Xn-1 ≠ j, Xn-2 ≠ j. … X1 ≠ j| X0 = i]
Now suppose we have Tij =Min {n :Xn = j | X0 = i}:
Thus we see if n=1 the, fij(1)= pij.
Then from the above expressions we get the result:
There is two type of states which we can get from the above result as:
Lastly we come to the need of taking out the average recurrence time i.e. the time taken by the state to return back to its initial position. The equation is:.
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