Discrete and Continuous Time Modulated Random Walks with Heavy-Tailed Increments

We consider a modulated process S which, conditional on a background process X, has independent increments. Assuming that S drifts to −∞ and that its increments (jumps) are heavy-tailed (in a sense made precise in the paper), we exhibit natural conditions under which the asymptotics of the tail distribution of the overall maximum of S can be computed. We present results in discrete and in continuous time. In particular, in the absence of modulation, the process S in continuous time reduces to a Lévy process with heavy-tailed Lévy measure. A central point of the paper is that we make full use of the so-called “principle of a single big jump” in order to obtain both upper and lower bounds. Thus, the proofs are entirely probabilistic. The paper is motivated by queueing and Lévy stochastic networks.     

Stabilizing an uncertain production system

Consider a production system that consists ofm machines each of which can produce parts ofn types. When machinek is used, it produces a part of typei with probabilityp _ki . Requests arrive for parts, one at a time. With probability _i an arriving request is for a part of typei. The requests must be served without waiting. Thus, if a requested part is not available, it must be produced. We find necessary and sufficient conditions for the existence of a strategy (a choice of the machines to be used) which makes the inventory of parts stable and we provide such a strategy.Two variations of this model are also considered: the case of batch arrivals of requests, and that of a system where the requests can be queued.     

Monitoring Phase II Comparative Clinical Trials with Two Endpoints and Penalty for Adverse Events

Adverse events in Phase II comparative clinical trials have received limited attention in the literature. Bersimis et al. (Stat Med 34:197–214, 2014) in proposed a class of comparative sequential designs with bivariate endpoints, where as a special case, the termination of the clinical trial due to the occurrence of a severe adverse event is treated. In this paper, using the Markov chain embedding technique, we extend this class of designs proposing two new designs, which treat cases where the development of an adverse event does not immediately stop the clinical trial, but penalizes appropriately the treatment that caused it. In both designs the penalty can be chosen either by assessing the severity of the adverse event or by optimizing the power. The numerical results show an excellent performance, achieving small expected sample sizes in conjunction with large values for power, satisfying in this way the ethical requirement for small sample sizes and fast decisions in clinical practice. The formulation of the procedure as a stochastic process is elegantly accomplished while it offers the necessary mathematical framework for further generalizing the designs covering more cases such as group sequential designs, etc.     

The distribution of age-of-information performance measures for message processing systems

Queueing Systems - The idea behind the recently introduced “age-of-information” performance measure of a network message processing system is that it indicates our knowledge regarding...     

New features in the class of association models

The most common association models are the U-, R-, C- and RC-models, achieved when only one dimension on the association is considered as significant (M = 1). Whenever more than one dimensions are significant (M > 1), the only kind of association assumed for each dimension is of the RC-type with the exception of Goodman's R + C + RC, R + RC, C + RC, U + RC and R + C models. Extending the idea of U-, R-, and C-models to association models with M > 1, relaxing the equidistance principle for successive categories for the known scores and using orthogonal polynomials, some new models are developed. Two classical sets of data are used to illustrate the advantages of the newly introduced models. Comparisons in terms of chi-square goodness of fit indicate that some of the newly introduced models perform better than the models fitted so far on these data sets. © 1998 John Wiley & Sons, Ltd.     

A multilinear algebra proof of the Cauchy–Binet formula and a multilinear version of Parsevalʼs identity

We prove the Cauchy–Binet determinantal formula using multilinear algebra by first generalizing it to an identity not involving determinants. By extending the formula to abstract Hilbert spaces we obtain, as a corollary, a generalization of the classical Parseval identity.     

Divergences without probability vectors and their applications

In general, divergences and measures of information are defined for probability vectors. However, in some cases, divergences are ‘informally’ used to measure the discrepancy between vectors, which are not necessarily probability vectors. In this paper we examine whether divergences with nonprobability vectors in their arguments share the properties of probabilistic or information theoretic divergences. The results indicate that divergences with nonprobability vectors share, under some conditions, some of the properties of probabilistic or information theoretic divergences and therefore can be considered and used as information measures. We then use these divergences in the problem of actuarial graduation of mortality rates. Copyright © 2009 John Wiley & Sons, Ltd.     

A fully stochastic approach to limit theorems for iterates of Bernstein operators

This paper presents a stochastic approach to theorems concerning the behavior of iterations of the Bernstein operator $B_n$ taking a continuous function $f\in C[0,1]$ to a degree-$n$ polynomial when the number of iterations $k$ tends to infinity and $n$ is kept fixed or when $n$ tends to infinity as well. In the first instance, the underlying stochastic process is the so-called Wright–Fisher model, whereas, in the second instance, the underlying stochastic process is the Wright–Fisher diffusion. Both processes are probably the most basic ones in mathematical genetics. By using Markov chain theory and stochastic compositions, we explain probabilistically a theorem due to Kelisky and Rivlin, and by using stochastic calculus we compute a formula for the application of $B_n$ a number of times $k=k\left(n\right)$ to a polynomial $f$ when $k\left(n\right)∕n$ tends to a constant.