Renewal Theory

To offer a view into the rapidly developing theory of fractional diffusion processes we describe in some detail three topics of present interest: (i) the well-scaled passage to the limit from continuous time random walk under power law assumptions to space-time fractional diffusion, (ii) the asymptotic universality of the Mittag-Leffler waiting time law in time-fractional processes, (iii) our method of parametric subordination for generating particle trajectories.

After sketching the basic principles of renewal theory we discuss the classical Poisson process and offer two other processes, namely the renewal process of Mittag-Leffler type and the renewal process of Wright type, so named by us because special functions of Mittag-Leffler and of Wright type appear in the definition of the relevant waiting times. We compare these three processes with each other, furthermore consider corresponding renewal processes with reward and numerically their long-time behaviour.

In this article we obtain rates of convergence to equilibrium of marked Hawkes processes in two situations. Firstly, the stationary process is the empty process, in which case we speak of the rate of extinction. Secondly, the stationary process is the unique stationary and nontrivial marked Hawkes process, in which case we speak of the rate of installation. The first situation models small epidemics, whereas the results in the second case are useful in deriving stopping rules for simulation algorithms of Hawkes processes with random marks.

It is our intention to provide via fractional calculus a generalization of the pure and compound Poisson processes, which are known to play a fundamental role in renewal theory, without and with reward, respectively. We first recall the basic renewal theory including its fundamental concepts like waiting time between events, the survival probability, the counting function. If the waiting time is exponentially distributed we have a Poisson process, which is Markovian. However, other waiting time distributions are also relevant in applications, in particular such ones with a fat tail caused by a power law decay of its density. In this context we analyze a non-Markovian renewal process with a waiting time distribution described by the Mittag-Leffler function. This distribution, containing the exponential as particular case, is shown to play a fundamental role in the infinite thinning procedure of a generic renewal process governed by a power asymptotic waiting time. We then consider the renewal theory with reward that implies a random walk subordinated to a renewal process.

The Austrian school faces a true ambiguity concerning its ability to develop a positive theory of the firm. Although it never really forms the subject of specific studies, the enterprise nevertheless implicitly appears both in Austrian theories of entrepreneurship—which deal with the market behavior of enterprises facing the problem of knowledge—and in their writings on the microeconomic analysis of the

In this paper, a flexible discrete-time arrival process is introduced and its correlation properties are analyzed. The arrival process is the so-called batch-on/off model, an extension of the original on/off source used in the context of ATM networks. In the batch-on/off model, a group of arrivals may be generated at any given active slot. General distributions are assumed for the

Diffusion and relaxation in many disordered systems exhibits anomalous behaviors, for example a mean square displacement of a tracer particle may follow < r2 > ~ talpha . The continuous time random walk (CTRW), introduced by Montroll and Weiss (1965), is a stochastic framework applied extensively to model such anomalous behaviors. In recent years it has become clear that anomalous diffusion processes may exhibit aging behavior. We investigate aging CTRW, previously introduced by Monthus and Bouchaud (1996) in the context of diffusion in glasses. Now the Green function P(r,ta,t) depends on the age of the diffusion process, instead of the usual Montroll--Weiss Green function P(r,t). An exact equation for the Fourier double Laplace transform of the aging CTRW Green function P(r,ta,t) is derived, thus generalizing the Montroll--Weiss equation to the aging regime. Asymptotic long time t and long time ta behaviors of aging CTRW are derived. Aging CTRW might be applied to any random walk process, described by standard CTRW, provided that aging initial conditions are satisfied. For example, recently, the approach was successfully applied to chaotic diffusion generated by a deterministic non-linear low dimensional system.

Over the past century the study of gas exchange rates between the atmosphere and the ocean has received increased attention because of concern about the fate of greenhouse gases such as CO2 released into the atmosphere. Of interest is the oceanic uptake of CO2 in shallow water coastal regions as biological productivity in these regions is on average about three times larger than in the open ocean. It is well-known that in the absence of breaking surface waves, the water side turbulence controls gas transfer of sparingly soluble gases such as CO2 from the air to the water. The dependence of gas transfer on winddriven shear turbulence and convection turbulence generated by surface cooling has been investigated previously by others. However, the effect of Langmuir turbulence generated by wave-current interaction has not been investigated before. More specifically, Langmuir turbulence is generated by the interaction of the wind-driven shear current with the Stokes drift velocity induced...

We consider the asymptotic variance of the departure counting processD(t) of the GI/G/1 queue;D(t) denotes the number of departures up to timet. We focus on the case where the system load ϱ equals 1, and prove that the asymptotic variance rate satisfies limt→∞varD(t) /t= λ(1–2/π)(ca2+cs2), where λ is the arrival rate, andca2andcs2are squared coefficients of variation of the interarrival and service times, respectively. As a consequence, the departures variability has a remarkable singularity in the case in which ϱ equals 1, in line with the BRAVO (balancing reduces asymptotic variance of outputs) effect which was previously encountered in finite-capacity birth-death queues. Under certain technical conditions, our result generalizes to multiserver queues, as well as to queues with more general arrival and service patterns. For the M/M/1 queue, we present an explicit expression of the variance ofD(t) for anyt.

This article compares the statistical properties of the records from independent and identically distributed time series with those of time series containing a single unit root. It is shown that there are important differences in both the limiting distributions and the convergence rates of the associated record counting processes. Since the record properties of independent and identically distributed time series are shared by a large class of stationary time series, the reported differences underline the possibility of using record-based statistics for robust testing procedures of the unit root hypothesis. We make an extension of the nonparametric test for the Range Unit Root test (RUR) proposed in Aparicio et al. (2006). We prove some properties for the test statistic in the context of the renewal theory and we suggest two new candidates to test the hypothesis of random walk with positive and negative drift.

In this paper the idea of treating the operational service workers as the elements of technique systems is suggested and the renewal theory is used to forecast the number of accidents caused by human factor. The analytical model is presented and limitations for its application are ...

The Monte Carlo EM (MCEM) algorithm is a modification of the EM algorithm where the expectation in the E-step is computed numerically through Monte Carlo simulations. The most flexible and generally applicable approach to obtaining a Monte Carlo sample in each iteration of an MCEM algorithm is through Markov chain Monte Carlo (MCMC) routines such as the Gibbs and Metropolis-Hastings samplers. Although MCMC estimation presents a tractable solution to problems where the E-step is not available in closed form, two issues arise when implementing this MCEM routine: (1) how do we minimize the computational cost in obtaining an MCMC sample? and (2) how do we choose the Monte Carlo sample size? We address the first question through an application of importance sampling whereby samples drawn during previous EM iterations are recycled rather than running an MCMC sampler each MCEM iteration. The second question is addressed through an application of regenerative simulation. We obtain approxima...

We consider an insurance portfolio situation in which there is possible dependence between the waiting time for a claim and its actual size. By employing the underlying random walk structure we obtain explicit exponential estimates for infinite- and finite-time ruin probabilities in the case of light-tailed claim sizes. The results are illustrated in several examples, worked out for specific dependence structures.