Background and Fundamental Results
In this chapter we present a number of fundamental results that will be used throughout the remainder of the book. The chapter is organized as follows. In Section 2.2 we collect some definitions and basic properties of deterministic point processes, and present two versions of Y = λX — the sample-path analogue of the renewal-reward theorem. (These generalize the simple version of Y = λX given in Chapter 1.) Section 2.3 presents fluid versions of Y = λX, in which the point process is replaced by a cumulative process. Section 2.4 shows how Y = λX can be used to give a simple proof of the sample-path rate-conservation law RCL under more general conditions than previously given. Section 2.5 gives discrete-variable and continuous-variable versions of the Fundamental Lemma of Maxima. In Section 2.6 we give a result that proves the equality of time-averages of a process and the mean of its asymptotic frequency distribution under a uniform integrability condition. In subsequent chapters we apply Y = λX and RCL to obtain simple proofs of relations between continuous-time frequencies of a process with a general state space and frequencies at the points of an imbedded point process.
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