Module 4:Renewal Processes and Theory, Limit theorems in renewal theory
Lecture 16:Proof of Renewal Theory
Proposition 4.4
Proof of Proposition 4.4
Now it is true that , such that we can also write it as . Now watch carefully the first term which is and it is the average of first inter arrival times and by the SLLN we have as . Now when , hence as .
Also write . Again using similar logic we can deduce that as and separately we note that . Utilizing these we have as .
Hence we have as each of the left and right hand bounds separately converge to as
.
, such that we can also write it as 
. Now watch carefully the first term which is 
and it is the average of first 
inter arrival times and by the SLLN we have 
as 
. Now 
when 
, hence 
as 
. Again using similar logic we can deduce that 
as 
and separately we note that 
. Utilizing these we have 
as 
.
as each of the left and right hand bounds separately converge to 
as
. 
