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Unformatted text preview: STAT 333 Applications of the Aperiodic Renewal Theorem The Aperiodic Renewal Theorem states that for an aperiodic renewal event λ : E ( T λ ) = 1 lim n →∞ r n where r n , n = 1 , 2 , . .. is the renewal sequence Examples: 1. Roll a fair die. Let λ =“2 4’. This is an aperiodic renewal event, r n = ( 1 6 )( 1 6 ) = 1 36 for n ≥ 2. Thus lim n →∞ r n = 1 36 and so E ( T λ ) = 36 . 2. Suppose digits are selected randomly with replacement from the set { , 1 , 2 , ..., 9 } . Let λ = “8 8 9 8 8 6”. This is an aperiodic renewal event with r n = 1 10 6 for n ≥ 6. Thus lim n →∞ r n = 1 10 6 and so E ( T λ ) = 10 6 = 1 , 000 , 000 . From the above two examples we can see that the application of the Renewal Theorem to such simple aperiodic renewal events is straightforward and almost boring (and a big improvement over computing generating functions or setting up recursive relations!) An additional particularly nice feature of the Renewal Theorem is that we can use it to compute...
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This note was uploaded on 08/01/2008 for the course STAT 333 taught by Professor Chisholm during the Spring '08 term at Waterloo.
 Spring '08
 Chisholm

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