10. Renewal Processes

This is the first chapter in the book to deal with random processes in continuous time, namely, with the so-called renewal processes. Section 10.1 establishes the basic terminology and proves the integral renewal theorem in the case of non-identically distributed random variables. The classical Key Renewal Theorem in the arithmetic case is proved in Sect. 10.2, including its extension to the case where random variables can assume negative values. The limiting behaviour of the excess and defect of a random walk at a growing level is established in Sect. 10.3. Then these results are extended to the non-arithmetic case in Sect. 10.4. Section 10.5 is devoted to the Law of Large Numbers and the Central Limit Theorem for renewal processes. It also contains the proofs of these laws for the maxima of sums of independent non-identically distributed random variables that can take values of both signs, and a local limit theorem for the first hitting time of a growing level. The chapter ends with Sect. 10.6 introducing generalised (compound) renewal processes and establishing for them the Central Limit Theorem, in both integral and integro-local forms.