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Register a new userRenewal series and square-root boundaries for Bessel processes
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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 motion, and recovers by different means, extensions for Bessel processes, obtained independently by Delong and Yor.
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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.
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 problem of universal estimators is to do so without any prior knowledge of the distribution. We prove a variety of results of this type, including universal estimates for the expected time to renewal as well as estimates for the conditional distribution of the time to renewal. Some of our results require a moment condition on the time to renewal and we show by an explicit construction how some moment condition is necessary.
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.
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 nonnegative reproduction function corresponds to self-excitation, and has been widely investigated in the literature. In particular, there exists a cluster representation of the Hawkes process which allows to apply results known for Galton-Watson trees. In the present paper, we establish limit theorems for Hawkes process with signed reproduction functions by using renewal techniques. We notably prove exponential concentration inequalities, and thus extend results of Reynaud-Bouret and Roy (2007) which were proved for nonnegative reproduction functions using this cluster representation which is no longer valid in our case. An important step for this is to establish the existence of exponential moments for renewal times of M/G/infinity queues that appear naturally in our problem. These results have their own interest, independently of the original problem for the Hawkes processes.
We consider the process ${x-N(t):tgeq 0}$, where $x>0$ and ${N(t):tgeq 0}$ is a renewal process with light-tailed distributed holding times. We are interested in the joint distribution of $(tau(x),A(x))$ where $tau(x)$ is the first-passage time of ${x-N(t):tgeq 0}$ to reach zero or a negative value, and $A(x)$ is the corresponding first-passage area. We remark that we can define the sequence ${(tau(n),A(n)):ngeq 1}$ by referring to the concept of integrated random walk. Our aim is to prove asymptotic results as $xtoinfty$ in the fashion of large (and moderate) deviations
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