Let F be the gamma distribution function with parameters a > 0 and α > 0 and let Gs be the negative binomial distribution function with parameters α and a/s, s > 0. By combining both probabilistic and approximation-theoretic methods, we obtain sharp upper and lower bounds for . In particular, we show that the exact order of uniform convergence is s–p, where p = min(1, α). Various kinds of applications concerning charged multiplicity distributions, the Yule birth process and Bernstein-type operators are also given.

In this paper, we develop the probabilistic foundations of the dynamic continuous choice problem. The underlying choice set is a compact metric space E such as the unit interval or the unit square. At each time point t, utilities for alternatives are given by a random function . To achieve a model of dynamic continuous choice, the theory of classical vector-valued extremal processes is extended to super-extremal processesY= {Yt, t > 0}. At any t > 0, Yt is a random upper semicontinuous function on a locally compact, separable, metric space E. General path properties of Y are discussed and it is shown that Y is Markov with state-space US(E). For each t > 0, Yt is associated.

For a compact metric E, we consider the argmax process M = {Mt, t > 0}, where . In the dynamic continuous choice application, the argmax process M represents the evolution of the set of random utility maximizing alternatives. M is a closed set-valued random process, and its path properties are investigated.

The diffusions on the shape and size-and-shape spaces induced by brownian motions on the pre-size-and-shape spaces have been investigated in several papers (cf.). We here address the dual problem: the character of the diffusions on the pre-shape and pre-size-and-shape spaces which induce brownian motions on the shape and size-and-shape spaces. In particular we show that the shape and size-and-shape spaces for k labelled points in ℝm are stochastically complete if k > m and obtain the heat kernels of certain diffusions which induce brownian motions on the size-and-shape spaces.

Statistical exponential families invariant with respect to the groups of the cone or the paraboloid of revolution are discussed. B(x, y) denotes the symmetric bilinear form on x0y0 – x1y1 – ·· ·– xdyd on ℝd+1, C denotes the cone of revolution in ℝd+1 {x; B(x,x) > 0 and x0 > 0}, and, for p > ½(d−1), μ p is the positive measure on ℝd+1 defined by its Laplace transform ⨍ exp (B(x,y))μp(dy) = (B(y,y))−p for y on C. More precisely, if p > ½ (d−1) one has and μ½(d−1) concentrated on the boundary ∂C. This paper studies the natural exponential families

Our aim in this article is to derive an expression for the best linear predictor of a multivariate symmetric α stable process based on many past values. For this purpose we introduce a definition of dispersion for symmetric α stable random vectors and choose the linear predictor which minimizes the dispersion of the error vector.

A proposal is given for estimating the home range of an animal based on sequential sightings. We assume the given sightings are independent, identically distributed random vectors X1,· ··, Xn whose common distribution has compact support. If are the polar coordinates of the sightings, then is a sup-measure and corresponds to the right endpoint of the distribution . The corresponding upper semi-continuous function l(θ) is the boundary of the home range. We give a consistent estimator for the boundary l and under the assumption that the distribution of R1 given is in the domain of attraction of an extreme value distribution with bounded support, we are able to give an approximate confidence region.

Hierarchical information networks are important in applications where the information management must support an existing tree-structured organization. Embedded computer-communication systems in military applications, with a dominant hierarchical command structure, are the most prominent examples. Also typical of such applications is the variability in message demand (both sources and intensity) depending on the external conditions encountered by the encapsulating system (the system supported by the embedded computer-communication system). Using a queueing network model for a hierarchical information network, we compare the effect of limited dynamic reconfiguration on expected transmission delays. The limited reconfigurability takes the form of apex transition among a proper subset of the communication nodes designated as the apex candidate set. Each apex candidate can assume the ultimate position under designated conditions. This network architecture is called a dynamic hierarchy. The model includes N + 1 nodes (0, …, N) with 0 identifying the apex node. We assume that message processing at each node is described by an M/M/1 model (single server with Poisson arrivals and exponential service times). Further message transfers among the nodes are served by communication links which also behave as M/M/1 queues.

Two distinctive features characterize the queueing network model:

1. The assignment of a set of weights to the nodes dependent on the hierarchical level reflects the increasing importance of information as it is transferred to higher levels.

2. The dynamic hierarchy requires a communications protocol that partitions the analysis of network delay into three periods: regular operation, reconfiguration, and adjustment. Characterization of the performance of the dynamic hierarchy entails the description of message transmission delay as a composite of the three periods.

We present two sufficient conditions for detection of optimal and non-optimal actions in (ergodic) average-cost MDPs. They are easily interpreted and can be implemented as detection tests in both policy iteration and linear programming methods. An efficient implementation of a recent new policy iteration scheme is discussed.

In the theory of autoregressive model fitting it is of interest to know the asymptotic behaviour, for large sample size, of the coefficients fitted. A significant role is played in this connection by the moments of the norms of the inverse sample covariance matrices. We establish uniform boundedness results for these, first under generally weak conditions and then for the special case of (infinite order) processes. These in turn imply corresponding ergodic theorems for the matrices in question.

We derive two kinds of rate conservation laws for describing the time-dependent behavior of a process defined with a stationary marked point process and starting at time 0. These formulas are called TRCLs (time-dependent rate conservation laws). It is shown that TRCLs are useful to study the transient behaviors of risk and storage processes with stationary claim and supply processes and with a general premium and release rates, respectively. Detailed discussions are given for the severity for the risk process, and for the workload process of a single-server queue.

Consider a number of events in a probability space. Let X be a random variable that is the number of events that occur. Given some of the moments of the distribution of X, it is possible to obtain bounds on the probability that at least one event occurs. The best possible bounds are given here. If there are many equiprobable events that are d- wise independent, and d is even, then the probability that at least one event happens is at least 1 — O(µ–d/2), where μ = E(X).

The concept of max-infinite divisibility is viewed as a positive dependence concept. It is shown that every max-infinitely divisible distribution function is a multivariate totally positive function of order 2 (MTP2). Inequalities are derived, with emphasis on exchangeable distributions. Applications and examples are given throughout the paper.

The expected number of upcrossings for a max-stable process is computed and compared with known results for stable processes. Asymptotically the formulas are of the same order.

Kalashnikov and Rachev (1986) have proposed a partial ordering of life distributions which is equivalent to an increasing hazard ratio, when the ratio exists. This model can represent the phenomenon of crossing hazards, which has received considerable attention in recent years. In this paper we study this and two other models of relative ageing. Their connections with common partial orderings in the reliability literature are discussed. We examine the closure properties of the three orderings under several operations. Finally, we give reliability and moment bounds for a distribution when it is ordered with respect to a known distribution.

In this contribution we consider an M/M/1 queueing model with general server vacations. Transient and steady state analysis are carried out in discrete time by combinatorial methods. Using weak convergence of discrete-parameter Markov chains we also obtain formulas for the corresponding continuous-time queueing model. As a special case we discuss briefly a queueing system with a T-policy operating.

Allocation of a redundant component in a system in order to optimize, in some sense, the lifetime of the system is an important problem in reliability theory, having practical applications. Consider a series system consisting of two components (say C1 and C2), having independent random lifetimes X1 and X2, and suppose a component C having random lifetime X (independent of X1 and X2) is available for active redundancy with one of the components. Let U1 = min(max(X1, X), X2) and U2 = min(X1, max(X2, X)), so that U1 (U2) denote the lifetime of a system obtained by allocating C to C1 (C2). We consider the criterion where C1 is preferred to C2 for redundancy allocation if . Here we investigate the problem of allocating C to C1 or C2, with respect to the above criterion. We also consider the standby redundancy for series and parallel systems with respect to the above criterion. The problem of allocating an active redundant component in order that the resulting system has the smallest failure rate function is also considered and it is observed that unlike stochastic optimization, here the lifetime distribution of the redundant component also plays a role, making the problem of even active redundancy allocation more complex.

We consider the standard single-server queue with unlimited waiting space and the first-in first-out service discipline, but without any explicit independence conditions on the interarrival and service times. We find conditions for the steady-state waiting-time distribution to have asymptotics of the form x–1 log P(W> x) → –θ ∗as x → ∞for θ ∗ > 0. We require only stationarity of the basic sequence of service times minus interarrival times and a Gärtner–Ellis condition for the cumulant generating function of the associated partial sums, i.e. n–1 log E exp (θSn) → ψ (θ) as n → ∞, plus regularity conditions on the decay rate function ψ. The asymptotic decay rate θ is the root of the equation ψ (θ) = 0. This result in turn implies a corresponding asymptotic result for the steady-state workload in a queue with general non-decreasing input. This asymptotic result covers the case of multiple independent sources, so that it provides additional theoretical support for a concept of effective bandwidths for admission control in multiclass queues based on asymptotic decay rates.

The aim of this paper is to extend the existing theory of second-order self-similar processes as defined by Cox (1984) from the univariate case to higher dimensions. Multivariate self-similar processes defined in terms of second-order theory for stationary time series can be used as models for long-range dependent observations when the marginal observations are long-range dependent. An interesting question concerns the correlation structure within the processes when the marginal processes are correlated. We show that the self-similarity requirement, as defined in this article, implies a cross-correlation structure similar to that for the marginal processes. This occurs both in the time domain and in the frequency domain. This fact can be used to obtain generalized least squares estimates for the long-range dependence parameters. We discuss some difficulties concerning estimation based on simulations.

In this paper we obtain the Beňes equation for the evolution of the probability distribution of the excursion process associated with the level crossings of a general storage process. We then show that under stationarity and ergodicity assumptions on the process we can recover the well-known rate conservation law (RCL). Using the stationary solution we then show that the existence of an invariant solution can be studied in terms of an operator equation and we show how this characterization leads to a very simple explicit computation of the stationary distribution.

We provide a unified approach for establishing even-moment bounds for partial sums for a class of weakly dependent random variables satisfying a stationarity condition. As applications, we discuss moment bounds for various types of mixing sequences. To obtain even-moment bounds, we use a ‘combinatorial argument' developed by Cox and Kim (1990).