This note gives the solution of the following problem concerning geometric probabilities. What is the probability p(Bd; 2) that the circumference determined by three points P, P1 and P2 chosen independently and uniformly at random in the interior of a d-dimensional unit ball Bd in Euclidean space Ed (d ≧ 2) is entirely contained in Bd? From our result we conclude that p(Bd; 2) →π /(3√3) as d →∞.

For a single-type Galton—Watson branching process with mean less than one and finite second moment, we establish an upper bound for the mean of the associated Yaglom limit. This bound is attained if and only if the generating function of the process is linear.

Let , be a recurrent Markov renewal process and Mik(t) be the expected value of Nk(t) provided that at the initial moment the system is in state i. It is shown that when the mean recurrence times μ ii are finite, the differences μ ij Mki (t) – t behave asymptotically the same for all states i and k.

A birth process is studied in which the birth rate at any time is a function of the difference between the current population size and a target corresponding to unit growth rate. If this controlling function is a decreasing function of its argument a stabilizing effect is to be expected. By supposing that the controlling function varies very slowly, series expansions for the properties of the process are obtained, the leading term corresponding to a diffusion approximation. The sequence of births considered as a point process of controlled variability is examined and approximations to the interval distribution and covariance density obtained.

Nelson [9], [10] has computed the generating function of return probabilities to the initial state for a particular Markov chain on permutations of three objects. The formula obtained is

The present paper studies three distinct Markov chains generalizing the Nelson chain: the so-called three-coloured chain, with some birth-and-death processes on ℤ as a particular case, a chain on a graph close to the graph of the edges of a cube, and the daisy library. Two other themes piece together these chains: the notion of coloured chain and the technique of computation by additive processes.

We consider networks with a very large or infinite number of nodes, linked by cable channels. The request which comes to a node is ordered to occupy a certain route of successive channels. The functioning of the system is regulated by the reserving of channels in order of the arrivals of the requests. Under some general conditions the existence of an ergodic region for such networks is proved. The practical value of the result lies in the fact that these conditions do not depend on the size of the graph.

A supplemented GSMP (generalized semi-Markov process) is a useful stochastic process for discussing fairly general queues including queueing networks. Although much work has been done on its insensitivity property, there are only a few papers on its general properties. This paper considers a supplemented GSMP in a general setting. Our main concern is with a system of Laplace–Stieltjes transforms of the steady state equations called the basic equations. The basic equations are derived directly under the stationary condition. It is shown that these basic equations with some other conditions characterize the stationary distribution. We mention how to get a solution to the basic equations when the solution is partially known or inferred. Their applications to queues are discussed.

In this article we introduce a simple tool to derive polynomial upper bounds for the probability of observing unusually large maximal components in some models of random graphs when considered at criticality. Specifically, we apply our method to a model of a random intersection graph, a random graph obtained through p-bond percolation on a general d-regular graph, and a model of an inhomogeneous random graph.

We consider an infinite Markov chain with states E0, E1, …, such that E1, E2, … is not closed, and for i ≧ 1 movement to the right is limited by one step. Simple algebraic characterizations are given for persistency of all states, and, if E0 is absorbing, simple expressions are given for the probabilities of staying forever among the transient states. Examples are furnished, and simple necessary conditions and sufficient conditions for the above characterizations are given.

Some relationships are derived between the asynchronous and partially synchronous counting and interval processes associated with a multivariate stationary point process. A few examples are given to illustrate some of these relationships.

It is shown that the celebrated result of Sparre Andersen for random walks and Lévy processes has intriguing consequences when the last time of the process in (-∞, 0], say σ, is added to the picture. In the case of no positive jumps this leads to six random times, all of which have the same distribution—the uniform distribution on [0, σ]. Surprisingly, this result does not appear in the literature, even though it is based on some classical observations concerning exchangeable increments.

General and unifying approaches are discussed for computing fundamental characteristics of both continuous-time and discrete-time birth-death processes. In particular, an exponential family framework is used to derive explicit expressions, in terms of continued fractions, for joint generating functions of first-passage times and a whole collection of associated random quantities, and a random sum representation is used to obtain formulae for means, variances and covariances of stopped reward functions defined on a birth-death process.

We construct Boltzmann processes using the formalism of random trees. We are then able to extend previous results about convergence toward the equilibrium law to interactions involving random numbers of particles. We even show a geometric rate of convergence for an extended class of processes, especially for those having a scaling invariant interaction mechanism.

We study a random record model where the observation Xi has continuous distribution function Fαi (αi > 0) and the number of available observations is random and independent of the observations. We obtain the joint distribution of the record values and inter-record times for our model. We investigate the distribution of the number of records when the number of observations has one of the common distributions and the α's increase geometrically or linearly. A particularly interesting case arises when the observations arrive at time points paced by a Poisson point process. For this model we obtain distributional results for the inter-arrival times of records for a large class of combinations of α structures and intensity functions.

We present a general recurrence model which provides a conceptual framework for well-known problems such as ascents, peaks, turning points, Bernstein's urn model, the Eggenberger–Pólya urn model and the hypergeometric distribution. Moreover, we show that the Frobenius-Harper technique, based on real roots of a generating function, can be applied to this general recurrence model (under simple conditions), and so a Berry–Esséen bound and local limit theorems can be found. This provides a simple and unified approach to asymptotic theory for diverse problems hitherto treated separately.

In this paper we study sums of micropulses that generate different kinds of processes. Fractional Brownian motion and bifractional Brownian motion are obtained as limit processes. Moreover, we not only prove the convergence of finite-dimensional laws but also, in some cases, convergence in distribution in the space of right-continuous functions with left limits. Finally, we obtain generalizations with multidimensional indices.

An M/G/1 service system with finite waiting room is studied. A customer is served by one server in two phases, during the first of which a place in the waiting room is occupied. Starting from a determinant expression for the Laplace-Stieltjes transform of the busy period given in [2] we obtain contour integral expressions for the transform and the expectation, thereby generalising a result by Cohen [1]. This is effected by developing the determinants along the first row.

It is well known that, in a Bienaymé-Galton–Watson process (Zn) with 1 < m = EZ1 < ∞ and EZ1 log Z1 <∞, the sequence of random variables Znm –n converges a.s. to a non–degenerate limit. When m =∞, an analogous result holds: for any 0< α < 1, it is possible to find functions U such that α n U (Zn) converges a.s. to a non-degenerate limit. In this paper, some sufficient conditions, expressed in terms of the probability generating function of Z1 and of its distribution function, are given under which a particular pair (α, U) is appropriate for (Zn). The most stringent set of conditions reduces, when U (x) x, to the requirements EZ1 = 1/α, EZ1 log Z1 <∞.

The Cramér–Rao inequality for the variance of an unbiassed estimator is first recalled, and the value of the ‘ideal' equation of estimation existing if the minimum bound is reached illustrated by examples. When this estimation equation is not available, the more general inequality due to Kiefer is more relevant. The analogous estimation equation corresponding to the attaining of the more general minimum bound is again illustrated by examples; and the general theory, which is also available for more than one parameter, is indicated for any location and/or scaling problem.