Theorem 4.7

In case for a renewal process  as

Proof 4.7

Step # 1

Suppose that  holds and also let us refer to the illustration given in Figure 4.6.

Figure 4.6: Schematic diagram to illustrate the concept asymptotic rate of progress of  and its effect on average rate of occurrence

If you see Figure 4.6 carefully then we know that . So let us take the expectations for which we have

i.e.,

Now we need to show that , so let us prove the less than inequality, i.e., .