Renewal and Regenerative Processes

Abstract

From the mathematical point of view, renewal theory is concerned with the renewal equation

$$ f(t) = g(t) + \int_{[0,t]} {f(t - s){\text{d}}F (s )} , $$

where F is the cumulative distribution function of a finite measure on the positive real line. Its main concern is the asymptotic behavior of the solution f (the existence and uniqueness of which is not a real issue under mild conditions, as we shall see). Once embedded in the framework of point processes, renewal theory, and in particular Blackwell’s theorem, becomes a fundamental tool of probability theory, particularly useful in the study of regenerative processes, a large and important class of stochastic processes which includes for instance the recurrent continuous-time HMCs and the semi-Markov processes.