 
Summary: Chapter 5
Renewal and Point processes
Not all stochastic processes are Markovian. In this chapter we will study a class of
processes called point processes. Ths chapter will seem quite dry and disconnected
from the previous material at first pass. However, it will play a critical role in the
study of stochastic models of population processes and, in particular, biochemical
reaction networks in later chapters. We start with a special class of point processes
called renewal processes.
5.1 Renewal Processes
A renewal process is used to model occurrences of events happening at random times,
where the times between the occurrences can be approximated by independent and
identically distributed random variables. These models are surprising useful as many
times even the most complicated models have within them an embedded renewal
process.
The formal model is as follows. We let Yn, n 1, be a sequence of independent
and identically distributed random variables which take only nonnegative values. We
also let Y0 be a nonnegative random variable, independent from Yn, n 1, though
not necessarily of the same distribution. The range of these random variables could be
discrete, perhaps {0, 1, 2, . . . }, or continuous, perhaps [0, ). The random variables
Yn will be the interevent times of the occurrences. We assume throughout that for
