collapsed" width="80%" style="text-align:left" !Proof |- |We may iterate the expectation about the first holding time: From the definition of the renewal process, we have So as required. |} Key renewal theorem Let X be a renewal process with renewal function and interrenewal mean . Let be a function satisfying: g is monotone and non-increasing The key renewal theorem states that, as : Renewal theorem Considering for any gives as a special case the renewal theorem: as The result can be proved using integral equations or by a coupling argument. Though a special case of the key renewal theorem, it can be used to deduce the full theorem, by considering step functions and then increasing sequences of step functions. Asymptotic properties Renewal processes and renewal-reward processes have properties analogous to the strong law of large numbers, which can be derived from the same theorem. If is a renewal process and is a renewal-reward process then: almost surely. {| class="toccolours collapsible collapsed" width="80%" style="text-align:left" !Proof |- |First consider . By definition we have: for all and so for all t ≥ 0. Now since we have: as almost surely (with probability 1). Hence: almost surely (using the strong law of large numbers); similarly: almost surely. Thus (since is sandwiched between the two terms) almost surely. Next consider . We have almost surely (using the first result and using the law of large numbers on ). |} Renewal processes additionally have a property analogous to the central limit theorem: Inspection paradox A curious feature of renewal processes is that if we wait some predetermined time t and then observe how large the renewal interval containing t is, we should expect it to be typically larger than a renewal interval of average size. Mathematically the inspection paradox states: for any t > 0 the renewal interval containing t is stochastically larger than the first renewal interval. That is, for all x > 0 and for all t > 0: where FS is the cumulative distribution function of the IID holding times Si. A vivid example is the bus waiting time paradox: For a given random distribution of bus arrivals, the average rider at a bus stop observes more delays than the average operator of the buses. The resolution of the paradox is that our sampled distribution at time t is size-biased (see sampling bias), in that the likelihood an interval is chosen is proportional