We present a uniqueness result for Gibbs point processes with interactions that come from a non-negative pair potential; in particular, we provide an explicit uniqueness region in terms of activity z and inverse temperature $\beta$ . The technique used relies on applying to the continuous setting the classical Dobrushin criterion. We also present a comparison to the two other uniqueness methods of cluster expansion and disagreement percolation, which can also be applied for this type of interaction.

A scale-free tree with the parameter β is very close to a star if β is just a bit larger than −1, whereas it is close to a random recursive tree if β is very large. Through the Zagreb index, we consider the whole scene of the evolution of the scale-free trees model as β goes from −1 to + ∞. The critical values of β are shown to be the first several nonnegative integer numbers. We get the first two moments and the asymptotic behaviors of this index of a scale-free tree for all β. The generalized plane-oriented recursive trees model is also mentioned in passing, as well as the Gordon-Scantlebury and the Platt indices, which are closely related to the Zagreb index.

Let a+ = a if a > 0, a+ = 0 otherwise, and a– = (– a)+.

Queueing systems are studied with a last-come, first-served queueing discipline and batch arrivals generated by a finite number of non-exponential sources. A closed-form expression is derived for the steady-state queue length distribution. This expression has a scaled geometric form and is insensitive to the input distribution. Moreover, an algorithm for the recursive computation of the normalizing constant and the busy source distribution is presented. The results are of both practical and theoretical interest as an extension of the standard Poisson batch input case.

In this paper, a fourth moment bound for partial sums of functionals of strongly ergodic Markov chains is established. This type of inequality plays an important role in the study of the empirical process invariance principle. This inequality is specially adapted to the technique of Dehling, Durieu, and Volný (2008). The same moment bound can be proved for dynamical systems whose transfer operator has some spectral properties. Examples of applications are given.

This paper introduces a new form of local balance and the corresponding product-form results. It is shown that these new product-form results allow capacity constraints at the stations of a queueing network without reversibility assumptions and without special blocking protocols. In particular, exact product-form results for heavily loaded queueing networks are obtained.

A solution is obtained for the signal-noise problem X = M + Z in which M and Z are independent Gaussian processes. Conditions on the processes are given which insure that the best estimate under generalized square — error loss is the conditional mean of M given X = x. A sequence of approximators of the best estimate is also given.

Several different orderings for the comparison of point processes have been introduced and their relationships discussed in Whitt [9], Daley [2] and Deng [4]. It is of some interest to know whether these orderings, in general, are preserved under various operations on point processes. Some results concerning limit operations were given in Deng [4].

In the present paper, we first further introduce some convex and concave orderings for counting processes, and survey the relationships among all orderings mentioned in [9], [4] and this paper. Then we focus our attention on the study of the conditions for the preservation of orderings under the operations of superposition, thinning, shift, and random change of time.

We prove a ratio identity giving a transform of the distribution of the position of a two barrier random walk as a ratio of the equivalent transforms for the one barrier random walk. This ratio identity holds for skip free random walks where the increments are dependent on both the previous increment and also the position of the random walk.

Let A1, A2, ···, An be events on a given probability space and let Br, n be the event that exactly r of the A's occur. Let further Sk (n) be the kth binomial moment of the number of the A's which occur. A sufficient condition is given for the existence of lim P (Br,n), as n→ + ∞, in terms of limits of the Sk(n)'s and a formula is given for the limit above. This formula for the limit is similar to the sieve theorem of Takács (1967) for infinite sequences of events and in the proof we make use of Takács's analytic method. The result is immediately applicable to the limit distribution of the maximum of (dependent) random variables X1, X2, ···, Xn by choosing Aj = {Xj ≧ x}. Our main theorem is reformulated for this special case and an example is given for illustration.

An approximate expression is found for the probability density of the recurrence time between major outbreaks of a host-vector disease such as yellow fever, in which the infectious period of a host is short and the lifetime of a vector is short compared with that of a host. The typical interval between epidemics based on the modal value of this distribution is obtained. These results are the analogues of those obtained by Bartlett (1956), (1966) for the periodicity of measles epidemics in small communities.

Consider a family of Boolean models, indexed by integers n≥1. The nth model features a Poisson point process in ℝn of intensity e{nρn}, and balls of independent and identically distributed radii distributed like X̅n√n. Assume that ρn→ρ as n→∞, and that X̅n satisfies a large deviations principle. We show that there then exist the three deterministic thresholds τd, the degree threshold, τp, the percolation probability threshold, and τv, the volume fraction threshold, such that, asymptotically as n tends to ∞, we have the following features. (i) For ρ<τd, almost every point is isolated, namely its ball intersects no other ball; (ii) for τd<ρ<τp, the mean number of balls intersected by a typical ball converges to ∞ and nevertheless there is no percolation; (iii) for τp<ρ<τv, the volume fraction is 0 and nevertheless percolation occurs; (iv) for τd<ρ<τv, the mean number of balls intersected by a typical ball converges to ∞ and nevertheless the volume fraction is 0; (v) for ρ>τv, the whole space is covered. The analysis of this asymptotic regime is motivated by problems in information theory, but it could be of independent interest in stochastic geometry. The relations between these three thresholds and the Shannon‒Poltyrev threshold are discussed.

We obtain a representation of an arbitrary discrete distribution with n mass points by an equiprobable mixture of r distributions, each of which has no more than a (≧2) mass points, where r is the smallest integer greater than or equal to (n – 1)/(a – 1). An application to the generation of discrete random variables on a computer is described, which has as an important special case Walker's (1977) alias method.

A target is assumed to move according to a Wiener process in ℝ1. The probability of detecting the target is computed in terms of the search effort which accumulates along the target's path. Under certain independence assumptions this probability is given by the expectation of an exponential functional of the process. It is shown in this note that the failure probability in a search for a Wiener target is asymptotically proportional to , where T is the accumulated time spent searching. The asymptotic failure probability is also shown to be independent of the position of the search in ℝ1. In a similar fashion, it is shown that the failure probability in a search for a Wiener target in ℝ2 is independent of the position of the search and asymptotically proportional to (c log T + l)–1, c >0

For a fixed k ∈ {1, …, d}, consider arbitrary random vectors X0, …, Xk ∈ ℝd such that the (k + 1)-tuples (UX0, …, UXk) have the same distribution for any rotation U. Let A be any nonsingular d × d matrix. We show that the k-dimensional volume of the convex hull of affinely transformed Xi satisfies \[|{\rm{conv}}(A{X_{\rm{0}}} \ldots ,A{X_k}){\rm{|}}\mathop {\rm{ = }}\limits^{\rm{D}} (|{P_\xi }\varepsilon |/{\kappa _k})|{\rm{conv}}\left( {{X_0}, \ldots ,{X_k}} \right)\], where ɛ:= {x ∈ ℝd : x┬ (A┬A)−1x ≤ 1} is an ellipsoid, Pξ denotes the orthogonal projection to a uniformly chosen random k-dimensional linear subspace ξ independent of X0, …, Xk, and κk is the volume of the unit k-dimensional ball. As an application, we derive the following integral geometry formula for ellipsoids: ck,d,p∫Ad,k |ɛ ∩ E|p+d+1μd,k(dE) = |ɛ|k+1∫Gd,k |PLɛ|pνd,k(dL), where $c_{k,d,p} = \big({\kappa_{d}^{k+1}}/{\kappa_k^{d+1}}\big) ({\kappa_{k(d+p)+k}}/{\kappa_{k(d+p)+d}})$. Here p > −1 and Ad,k and Gd,k are the affine and the linear Grassmannians equipped with their respective Haar measures. The p = 0 case reduces to an affine version of the integral formula of Furstenberg and Tzkoni (1971).

The product moment equations previously derived are first discussed for the infinite one- and two-dimensional Ising (or autologistic) model in the case of non-zero mean, as a prelude to an examination of the probability structure in the higher-dimensional (and nominally zero mean) case below the ‘critical temperature’. Of two simple possible models, A and B, both consistent with the division of the product moment p into ergodic, and long-range non-ergodic, components, such that ρ = r (1 – m2) + m2, where r is the intrinsic correlation coefficient, it is shown that the second model B appears appropriate to the three-dimensional ‘spherical model’, but the first model A to the Ising model. Model A is defined by xi = yi + M, where M = +m or –m, and E{yi} = 0; and Model B by

We consider the Klimov model for an open network of two types of jobs. Jobs of type i arrive at station i, have processing times that are exponentially distributed with parameter µi, and when processed either go on to station j with probability pij, or depart the network with probability pi0. Costs are charged at a rate that depends on the number of jobs of the two types in the system. It is shown that for arbitrary arrival processes the policy that gives priority to those jobs for whom the rate of change of the cost function is greatest minimizes the expected cost rate at every time t. This result is stronger than the Klimov result in two ways: arrival processes are arbitrary, and the minimization is at each time t. But the result holds for only two types.

Based on infinite and finite line of descent of particles of a one-dimensional supercritical Galton-Watson branching process (GWBP), we construct an associated bivariate process. We show that this bivariate process is a two-dimensional, supercritical GWBP. We also show that this process retains its branching property on appropriate probability spaces, when conditioned on set of non-extinction and set of extinction. Some asymptotic and weak convergence results for this process have been established. A generalisation of these results to a multitype p-dimensional GWBP has also been carried out.

The substability and the ergodicity of various queueing models are discussed. This paper considers the vector-valued queueing process Xrn = (xrn,1,xrn,2, · ··) with non-negative components and a constant initial value Xrr= a. For this, the substability is derived under simple conditions by showing the finiteness of With respect to the ergodicity, in order to make use of Borovkov's theorem, we additionally assume that the distribution of the interarrival of customers has non-bounded tail for any given past sequence.

The response of the photographic grain to light is a non-deterministic process which is as yet not completely understood. This response, as measured by the photographic density, is usually taken to be a function of the product of incident light intensity and exposure time interval duration, but at extreme values of either of these two quantities, this is no longer true. This latter effect is called reciprocity-law failure.

This paper discusses a probabilistic model, similar to a multiserver queue, for high-intensity reciprocity failure.