Probabilistic sampling of finite renewal processes

Consider a finite renewal process in the sense that interrenewal times are positive i.i.d. variables and the total number of renewals is a random variable, independent of interrenewal times. A finite point process can be obtained by probabilistic sampling of the finite renewal process, where each renewal is sampled with a fixed probability and independently of other renewals. The problem addressed in this work concerns statistical inference of the original distributions of the total number of renewals and interrenewal times from a sample of i.i.d. finite point processes obtained by sampling finite renewal processes. This problem is motivated by traffic measurements in the Internet in order to characterize flows of packets (which can be seen as finite renewal processes) and where the use of packet sampling is becoming prevalent due to increasing link speeds and limited storage and processing capacities.Comment: Published in at http://dx.doi.org/10.3150/10-BEJ321 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

On universal estimates for binary renewal processes

A binary renewal process is a stochastic process $\{X_n\}$ taking values in $\{0,1\}$ where the lengths of the runs of 1's 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.Comment: Published in at http://dx.doi.org/10.1214/07-AAP512 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Thickened renewal processes

AbstractLet {ti, i ⩾ 0} be an ordinary renewal process and assume the lifetime distribution function has the form F(x) = cxα (1 + λx + o(x)) near 0+. The asymptotic conditional distribution as n→∞ of {nti, i⩾0}, given that tn ⩽ 1, is that of a renewal process with a gamma lifetime distribution depending only on α

Quantum renewal processes

We introduce a general construction of master equations with memory kernel whose solutions are given by completely positive trace preserving maps. These dynamics going beyond the Lindblad paradigm are obtained with reference to classical renewal processes, so that they are termed quantum renewal processes. They can be described by means of semigroup dynamics interrupted by jumps, separated by independently distributed time intervals, following suitable waiting time distributions. In this framework, one can further introduce modified processes, in which the first few events follow different distributions. A crucial role, marking an important difference with respect to the classical case, is played by operator ordering. Indeed, for the same choice of basic quantum transformations, different quantum dynamics arise. In particular, for the case of modified processes, it is natural to consider the time inverted operator ordering, in which the last few events are distributed differently.Comment: 23 pages, 6 figure