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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 spatial dimension $d geq 1$ and requires only a moment condition slightly stronger than finite first moment. We also prove a Complete Convergence Theorem for heavy tailed interarrival times. Finally, for heavy tailed distributions we examine when the contact process, conditioned on survival, can be asymptotically predicted knowing the renewal processes. We close with an example of an interarrival time distribution attracted to a stable law of index 1 for which the critical value vanishes, a tail condition uncovered by previous results.
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 rene
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
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