This preview shows page 1. Sign up to view the full content.
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...
View
Full
Document
This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell University (Engineering School).
 Spring '10
 DURRETT
 The Land

Click to edit the document details