Module 4:Renewal Processes and Theory, Limit theorems in renewal theory
Proposition 4.3 for all , where is the renewal function.
Proof of Proposition 4.3 Since , then there must be some arbitrary , such that . Define another renewal process , such that and also let
Assume we have the following illustration, Figure 4.3 for Proposition 4.3
Figure 4.3: Schematic diagram to illustrate the concept of a renewal function and for the proof of Proposition 4.3
and suppose we can also write the following
For if , then , else For if , then , else . . For if , then , else
for all 
, where 
is the renewal function. 
, then there must be some arbitrary 
, such that 
. Define another renewal process 
, such that 
and also let 
if 
, then 
, else 
if 
, then 
, else 
if 
, then 
, else 
