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We review the theory of renewal reward processes, which describes renewal processes that have some cost or reward associated with each cycle. We present a new simplified proof of the renewal reward theorem that mimics the proof of the elementary renewal theorem and avoids the technicalities in the proof that is presented in most textbooks. Moreover, we mention briefly the extension of the theory to partial rewards, where it is assumed that rewards are not accrued only at renewal epochs but also during the renewal cycle. For this case, we present a counterexample which indicates that the standard conditions for the renewal reward theorem are not sufficient; additional regularity assumptions are necessary. We present a few examples to indicate the usefulness of this theory, where we prove the inspection paradox and Littles law through the renewal reward theorem.
This paper investigates Hawkes processes on the positive real line exhibiting both self-excitation and inhibition. Each point of this point process impacts its future intensity by the addition of a signed reproduction function. The case of a nonnegat
We consider the problem of determining escape probabilities from an interval of a general compound renewal process with drift. This problem is reduced to the solution of a certain integral equation. In an actuarial situation where only negative jumps
We refine some previous results concerning the Renewal Contact Processes. We significantly widen the family of distributions for the interarrival times for which the critical value can be shown to be strictly positive. The result now holds for any sp
A binary renewal process is a stochastic process ${X_n}$ taking values in ${0,1}$ where the lengths of the runs of 1s between successive zeros are independent. After observing ${X_0,X_1,...,X_n}$ one would like to predict the future behavior, and the
We show how a description of Brownian exponential functionals as a renewal series gives access to the law of the hitting time of a square-root boundary by a Bessel process. This extends classical results by Breiman and Shepp, concerning Brownian moti