4 for alternating renewal processes other

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Unformatted text preview: e theorem now gives: Theorem 3.8. Suppose Et1 < 1 and the greatest common divisor of {k : fk > 0} is 1 then P (t1 > i) lim P (Zn = i) = n!1 Et1 In particular P (Zn = 0) ! 1/Et1 . Example 3.9. Visits to Go. In Monopoly one rolls two dice and then moves that number of squares. As in Example 1.27 we will ignore Go to Jail, Chance, and other squares that make the chain complicated. The average number of spaces moved in one roll is Et1 = 7 so in the long run we land exactly on Go in 1/7 of the trips around the board. Using Theorem 3.8 we can calculate the limiting distribution of the amount we overshoot Go. 0 3.3.2 1 2 3 4 5 6 7 8 9 10 11 1 7 1 7 35 252 33 252 30 252 26 252 21 252 15 252 10 252 6 252 3 252 1 252 General case With the discrete case taken care of, we will proceed to the general case, which will be studied using renewal reward processes. Theorem 3.9. As t ! 1 Z Z1 1t 1 P (ti > z ) dz 1{As >x,Zs >y} ds ! t0 Et1 x+y Proof. Let Ix,y (s) = 1 if As > x and Zs &g...
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell University (Engineering School).

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