Probability and Stochastic Processes

11.3 The Key Renewal Theorem

This is the main result used in applications of renewal processes. We will start with a definition.

Definition 11.15 (Directly Riemann Integrable function)

A function is called a directly Riemann integrable (DRI) function if the upper and lower mesh Darboux sums are finite and have the same limit as .

Reminder of lower (and upper) Darboux sum LDS (and UDS):

Let be a partition of , with . Define

Example 11.6 of Riemann integrable function which is not DRI

Let

To understand this example, plot it from to some large number. We have that

so this function ...