Renewal Theory and Sharp Estimates

Proof of Theorem 2.2(1). Let us choose / such that f(/ ) 0. For the constrained case, the result is a direct consequence of the Renewal Theorem (Theorem A.3) and (2.18). For the free case we use the result of the constrained case to write 

The second statement follows by arguing as follows. First we write 

For Qjv we apply the Dominated Convergence Theorem by first observing that limjv-too 3C(N — n)/K N) = 1 for every n (this follows immediately 

We are left with showing that i jv = o(l). This can be done for a 0 by applying the rough bound L.l of Appendix A.4 and the first of the two 

Proof of Theorem 2.2(3). In this case (2.18) says that is the renewal 

---