Note

• Since the inter arrival times are i.i.d, hence at each renewal or event the process probability starts over again.
• There are finite numbers of renewals which can occur in a finite time, as by the strong law of large numbers (SLLN) we can show that . But since , hence  must go towards infinity as  goes to infinity. Thus,  can be less than or equal to  for at most a finite number of values of , hence  must be finite and one can write

For a better understanding of the concept of renewal theory it is important that we find the distribution of , but before that one must note the important relationship that the number of renewals by time  is greater than or equal to  iff the  renewal occurs before or at time . Hence we need to check the following theorem.

Theorem 4.1

Proof 4.1

                                                             

i.e.,

                        

Now ,  are i.i.d, hence it follows that  is distributed  which is the -fold convolution of itself which is the distribution of , .

Hence