This paper introduces a fractional heat equation, where the diffusion operator is the composition of the Bessel and Riesz potentials. Sharp bounds are obtained for the variance of the spatial and temporal increments of the solution. These bounds establish the degree of singularity of the sample paths of the solution. In the case of unbounded spatial domain, a solution is formulated in terms of the Fourier transform of its spatially and temporally homogeneous Green function. The spectral density of the resulting solution is then obtained explicitly. The result implies that the solution of the fractional heat equation may possess spatial long-range dependence asymptotically.

Statistics denoting the numbers of success runs of length exactly equal and at least equal to a fixed length, as well as the sum of the lengths of success runs of length greater than or equal to a specific length, are considered. They are defined on both linearly and circularly ordered binary sequences, derived according to the Pólya-Eggenberger urn model. A waiting time associated with the sum of lengths statistic in linear sequences is also examined. Exact marginal and joint probability distribution functions are obtained in terms of binomial coefficients by a simple unified combinatorial approach. Mean values are also derived in closed form. Computationally tractable formulae for conditional distributions, given the number of successes in the sequence, useful in nonparametric tests of randomness, are provided. The distribution of the length of the longest success run and the reliability of certain consecutive systems are deduced using specific probabilities of the studied statistics. Numerical examples are given to illustrate the theoretical results.

Among the disks centered at a typical particle of the two-dimensionalPoisson-Voronoi tessellation, let R m be the radius of the largest included within the polygonal cell associated with that particle and R M be the radius of the smallest containing that polygonal cell. In this article, we obtain the joint distribution of R m and R M. This result is derived from the covering properties of the circle due toStevens, Siegel and Holst. The same method works for studying the Crofton cell associated with the Poisson line process in the plane. The computation of the conditional probabilitiesP{R M ≥ r + s | R m = r}reveals the circular property of the Poisson-Voronoi typical cells(as well as the Crofton cells) having a ‘large’ in-disk.

In this last part the F n(i) and M n(i) are considered as random variables whose distributions are described by the model and various mating rules of Section 2. Several convergence results will be proved for those specific mating rules, but we begin with the more general convergence theorem 6.1. The proof of this theorem brings out the basic idea of this section, namely that when F n and M n are large, F n + 1(i) and M n + 1(i) will, with high probability, be close to a certain function of F n(·) and M n(·) (roughly the conditional expectation of F n+1(i) and M n + 1(i) given F n(·) and M n(·)). As we already indicated in Section 2, this leads (outside the exceptional set) to the approximate equality for some transformation T of the form (1.4), (1.5). More generally for fixed k except on a set whose probability is small when F n and M n are large. If the theorems of Section 3 or 4 apply, Tk (f n(·), m n(·)) will be close to a fixed vector ζ when k is large and thus there is hope that f n(·) and m n(·) will converge, once F n and M n become large. We therefore have to put on some conditions which will make F n and M n grow. This is the role of (6.34) and, to some extent, also of (6.17). The main difficulty is that the expected size of the (n + 1)th generation, given the nth generation, depends on the frequencies of the different types present in the nth generation. Even if (6.34) holds, the conditional expected size of the (n + 1)th generation, given the nth generation, may actually be smaller than the size of the nth generation for certain directions f n(·), m(·).

The signature of a system is defined as the vector whose ith element is the probability that the system fails concurrently with the ith component failure. The signature vector is known to be a distribution-free measure and a representation of the system's survival function has been developed in terms of the system's signature. The present work is devoted to the study of the joint distribution of lifetimes of pairs of systems with shared components. Here, a new distribution-free measure, the ‘joint bivariate signature’, of a pair of systems with shared components is defined, and a new representation theorem for the joint survival function of the system lifetimes is established. The theorem is shown to facilitate the study of the dependence between systems and the comparative performance of two pairs of such systems.

Kennedy (1972) showed that the standard single-server queueing model is continuous. These results are extended to the standard multi-server model here. Even when there is only one server, an additional condition is needed for the queue length process.

This paper discusses a number of explicit formulas for the steady-state features of the queue with Poisson arrivals in groups of random sizes and semi-Markovian service times. Computationally useful formulas for the expected duration of the various busy periods, for the mean numbers of customers served during them, as well as for the lower order moments of the queue lengths, both in discrete and in continuous time, and of the virtual waiting time are obtained. The formulas are recursive matrix expressions, which generalize the analogous but much simpler results for the classical M/G/1 model.

For several pairs (P, Q) of classical distributions on ℕ0, we show that their stochastic ordering P ≤stQ can be characterized by their extreme tail ordering equivalent to P({k *})/Q({k *}) ≥ 1 ≥ limk→k* P({k})/Q({k}), with k * and k * denoting the minimum and the supremum of the support of P + Q, and with the limit to be read as P({k *})/Q({k *}) for finite k *. This includes in particular all pairs where P and Q are both binomial (b n1,p1 ≤stb n2,p2 if and only if n 1 ≤ n 2 and (1 - p 1)n1 ≥ (1 - p 2)n2 , or p 1 = 0), both negative binomial (b − r1,p1 ≤stb − r2,p2 if and only if p 1 ≥ p 2 and p 1 r1 ≥ p 2 r2 ), or both hypergeometric with the same sample size parameter. The binomial case is contained in a known result about Bernoulli convolutions, the other two cases appear to be new. The emphasis of this paper is on providing a variety of different methods of proofs: (i) half monotone likelihood ratios, (ii) explicit coupling, (iii) Markov chain comparison, (iv) analytic calculation, and (v) comparison of Lévy measures. We give four proofs in the binomial case (methods (i)-(iv)) and three in the negative binomial case (methods (i), (iv), and (v)). The statement for hypergeometric distributions is proved via method (i).

In an earlier paper (Renewal theory in two dimensions: Basic results) the author developed a unified theory for the study of bivariate renewal processes. In contrast to this aforementioned work where explicit expressions were obtained, we develop some asymptotic results concerning the joint distribution of the bivariate renewal counting process (N x(1), N y(2)), the distribution of the two-dimensional renewal counting process N x,y and the two-dimensional renewal function &N x,y. A by-product of the investigation is the study of the distribution and moments of the minimum of two correlated normal random variables. A comprehensive bibliography on multi-dimensional renewal theory is also appended.

Inference of evolutionary trees and rates from biological sequences is commonly performed using continuous-time Markov models of character change. The Markov process evolves along an unknown tree while observations arise only from the tips of the tree. Rate heterogeneity is present in most real data sets and is accounted for by the use of flexible mixture models where each site is allowed its own rate. Very little has been rigorously established concerning the identifiability of the models currently in common use in data analysis, although nonidentifiability was proven for a semiparametric model and an incorrect proof of identifiability was published for a general parametric model (GTR + Γ + I). Here we prove that one of the most widely used models (GTR + Γ) is identifiable for generic parameters, and for all parameter choices in the case of four-state (DNA) models. This is the first proof of identifiability of a phylogenetic model with a continuous distribution of rates.

We consider the optimal stochastic scheduling of a two-stage tandem queue with two parallel servers. The servers can serve either queue at any point in time and the objective is to minimize the total holding costs incurred until all jobs leave the system. We characterize sufficient and necessary conditions under which it is optimal to allocate both servers to the upstream or downstream queue. We then conduct a numerical study to investigate whether the results shown for the static case also hold for the dynamic case. Finally, we provide a numerical study that explores the benefits of having two flexible parallel servers which can work at either queue versus servers dedicated to each queue. We discuss the results' implications for cross-training workers to perform multiple tasks.

We prove that the Heston volatility is Malliavin differentiable under the classical Novikov condition and give an explicit expression for the derivative. This result guarantees the applicability of Malliavin calculus in the framework of the Heston stochastic volatility model. Furthermore, we derive conditions on the parameters which assure the existence of the second Malliavin derivative of the Heston volatility. This allows us to apply recent results of Alòs (2006) in order to derive approximate option pricing formulae in the context of the Heston model. Numerical results are given.

By expressing the discounted net loss process as a randomly weighted sum, we investigate the finite-time ruin probabilities for the Poisson risk model with an exponential Lévy process investment return and heavy-tailed claims. It is found that in finite time, however, the extreme of insurance risk dominates the extreme of financial risk, but, for the case of dangerous investment (see Klüppelberg and Kostadinova (2008) for an accurate definition of dangerous investment), the extreme of financial risk has more and more of an effect on the total risk, and as time passes, the extreme of financial risk finally dominates the extreme of insurance risk.

We study Taylor series expansions of stationary characteristics of general-state-space Markov chains. The elements of the Taylor series are explicitly calculated and a lower bound for the radius of convergence of the Taylor series is established. The analysis provided in this paper applies to the case where the stationary characteristic is given through an unbounded sample performance function such as the second moment of the stationary waiting time in a queueing system.

We investigate the stability problem for a nonlinear autoregressive model with Markov switching. First we give conditions for the existence and the uniqueness of a stationary ergodic solution. The existence of moments of such a solution is then examined and we establish a strong law of large numbers for a wide class of unbounded functions, as well as a central limit theorem under an irreducibility condition.

We study a fork-join processing network in which jobs arrive according to a Poisson process and each job splits into m tasks, which are simultaneously assigned to m nodes that operate like M/M/s queueing systems. When all of its tasks are finished, the job is completed. The main result is a closed-form formula for approximating the distribution of the network's response time (the time to complete a job) in equilibrium. We also present an analogous approximation for the distribution of the equilibrium queue length (the number of jobs in the system), when each node has one server. Kolmogorov-Smirnov statistical tests show that these formulae are good fits for the distributions obtained from simulations.

Max-stable random fields play a central role in modeling extreme value phenomena. We obtain an explicit formula for the conditional probability in general max-linear models, which include a large class of max-stable random fields. As a consequence, we develop an algorithm for efficient and exact sampling from the conditional distributions. Our method provides a computational solution to the prediction problem for spectrally discrete max-stable random fields. This work offers new tools and a new perspective to many statistical inference problems for spatial extremes, arising, for example, in meteorology, geology, and environmental applications.

The paper deals with the asymptotic behaviour of infinite mean Galton–Watson processes (denoted by {Zn }). We show that these processes can be classified as regular or irregular. The regular ones are characterized by the property that for any sequence of positive constants {Cn }, for which a.s. exists, The irregular ones, which will be shown by examples to exist, have the property that there exists a sequence of constants {Cn } such that In Part 1 we study the properties of {Zn /Cn } and give some characterizations for both regular and irregular processes. Part 2 starts with an a.s. convergence result for {yn (Zn )}, where {yn } is a suitable chosen sequence of functions related to {Zn }. Using this, we then derive necessary and sufficient conditions for the a.s. convergence of {U(Zn )/Cn }, where U is a slowly varying function. The distribution function of the limit is shown to satisfy a Poincaré functional equation. Finally we show that for every process {Zn } it is possible to construct explicitly functions U, such that U(Zn )/en converges a.s. to a non-degenerate proper random variable. If the process is regular, all these functions U are slowly varying. The distribution of the limit depends on U, and we show that by appropriate choice of U we may get a limit distribution which has a positive and continuous density or is continuous but not absolutely continuous or even has no probability mass on certain intervals. This situation contrasts strongly with the finite mean case.

We consider a semi-Markov process with finite state space, partitioned into two classes termed ‘open' and ‘closed'. It is possible to observe only which class the process is in. We show that complete information concerning the aggregated process is contained in an embedded Markov renewal process, whose parameters, moments and equilibrium behaviour are determined. Such processes have found considerable application in stochastic modelling of single ion channels. In that setting there is time interval omission, i.e. brief sojourns in either class failed to be detected. Complete information on the aggregated process incorporating time interval omission is contained in a Markov renewal process, whose properties are derived, obtained from the above Markov renewal process by a further embedding. The embedded Markov renewal framework is natural, and its invariance to time interval omission leads to considerable economy in the derivation of properties of the observed process. The results are specialised to the case when the underlying process is a continuous-time Markov chain.

In this paper we consider a class of single-server queueing systems with compound Poisson arrivals, in which, at service completion epochs, the server has the option of taking off for one or several vacations of random length. The cost structure consists of a holding cost rate specified by a general non-decreasing function of the queue size, fixed costs for initiating and terminating service, and a variable operating cost incurred for each unit of time that the system is in operation. We show under some weak conditions with respect to the holding cost rate function and the service time, vacation time and arrival batch size distributions that it is either optimal among all feasible (stationary and non-stationary) policies never to take a vacation, or it is optimal to take a vacation when the system empties out and to resume work when, upon completion of a vacation, the queue size is equal to or in excess of a critical threshold. These optimality results are generalized for several variants of this model.