The size order problem of the probability state vector elements of a homogeneous Markov system is examined. The time t0 is evaluated, after which the order of the state vector probabilities remains unchanged, and a formula to quickly find a lower bound for t0 is given. A formula for approximating the mode of the state sizes ni(t) as a function of the means Eni(t), and a relation to evaluate P(ni(t) = x+1) by means of certain terms which constitute P(ni(t) = x) are derived.

We study a simple first-order nonnegative bilinear time-series model and give conditions under which the model is stationary. The probability density function of the stationary distribution (when it exists) is found. We also discuss the tail behaviour of the stationary distribution and calculate the probability density function by a numerical method. Simulation is used to check the calculation.

Let us consider n stocks with dependent price processes each following a geometric Brownian motion. We want to investigate the American perpetual put on an index of those stocks. We will provide inner and outer boundaries for its early exercise region by using a decomposition technique for optimal stopping.

This paper considers how students might be taught statistical consulting. Desirable characteristics of a good consultant are described. While skill in statistical techniques is considered, it is just one of several areas of expertise that are discussed. Suggestions are made as to what should be included in a consulting course to equip a student with these characteristics.

We present a new inspection policy useful when testing is needed to detect failures of a single-unit system. It is supposed that tests may fail and give an erroneous result. The inspection policy minimizing cost per unit of time for an infinite time span is also discussed. In addition, we study the behaviour of the optimum policy for some time to failure distributions often assumed in reliability: exponential and Pareto.

The assumption of linear costs of observation usually leads to optimal stopping boundaries which are straight lines. For non-linear costs of observation, the question arises of how the shape of cost functions influences the shape of optimal stopping boundaries. In Irle (1987), (1988) it was shown that, under suitable assumptions on c, for the problem of optimal stopping (Wt + x)+ - c(s + t), t ∊ [0,∞), the optimal stopping boundary h(t) can be enscribed between k1/c'(t) and k2/c'(t) for some constants k1, k2. In this paper we find the exact asymptotic expansion h(t) = 1/(4c'(t))(1 + o(1)).

For most repairable systems, the number N(t) of failed components at time t appears to be a good quality parameter, so it is critical to study this random function. Here the components are assumed to be independent and both their lifetime and their repair time are exponentially distributed. Moreover, the system is considered new at time 0. Our aim is to compare the random variable N(t) with N(∞), especially in terms of total variation distance. This analysis is used to prove a cut-off phenomenon in the same way as Ycart (1999) but without the assumption of identical components.

The aim of the present paper is to construct error correcting quantum codes from classical error correcting group codes by using the Schur orthogonality relations for characters of a finite abelian group and the Knill–Laflamme criterion for a quantum code to correct a pre-assigned family of errors. This is an abstraction and generalization of Shor's example of a nine qubit single error correcting quantum code.

We consider a controlled M/M/1 queueing system where customers may be subject to two potential rejections. The first occurs upon arrival and is dependent on the number of customers in the queue and the service rate of the customer currently in service. The second, which may or may not occur, occurs immediately prior to the customer receiving service. That is, after each service completion the customer in the front of the queue is assessed and the service rate of that customer is revealed. If the second decision-maker recommends rejection, the customer is denied service with a fixed probability. We show the existence of long-run average optimal monotone switching-curve policies. Further, we show that the average reward is increasing in the probability that the second decision-maker's recommendation of rejection is honored. Applications include call centers with delayed classifications and manufacturing systems when the server is responsible for multiple tasks.

The optimal stopping value of random variables X1,…,Xn depends on the joint distribution function of the random variables and hence on their marginals as well as on their dependence structure. The maximal and minimal values of the optimal stopping problem is determined within the class of all joint distributions with fixed marginals F1,…,Fn. They correspond to some sort of strong negative or positive dependence of the random variables. Any value inbetween these two extremes is attained for some dependence structures. The determination of the minimal value is based on some new ordering results for probability measures, in particular on lattice properties of stochastic orderings. We also identify properties of dependence structures leading to the minimal optimal stopping value. In the proofs we need an extension of Strassen's theorem on representation of the convex order which reveals that convex ordered distributions can be coupled by a two-step martingale (X,Y) with the additional property that Y is stochastically increasing in X.

We use multi-type branching process theory to construct a cell population model, general enough to include a large class of such models, and we use an abstract version of the Perron-Frobenius theorem to prove the existence of the stable birth-type distribution. The generality of the model implies that a stable birth-size distribution exists in most size-structured cell cycle models. By adding the assumption of a critical size that each cell has to pass before division, called the nonoverlapping case, we get an explicit analytical expression for the stable birth-type distribution.

In this paper, we consider the stochastic sequence {Yt}t∊ℕ defined recursively by the linear relation Yt+1 = AtYt + Bt in a random environment which is described by the non-stationary process {(At, Bt)}t∊ℕ. We formulate sufficient conditions on the environment which ensure that the finite-dimensional distributions of {Yt}t∊ℕ converge weakly to the finite-dimensional distributions of a unique stationary process. If the driving sequence {(At, Bt)}t∊ℕ becomes stationary in the long run, then we can establish a global convergence result. This extends results of Brandt (1986) and Borovkov (1998) from the stationary to the non-stationary case.

We consider a multicomponent maintenance system controlled by an age replacement policy: when one of the components fails, it is immediately replaced; all components older than a threshold age θ are preventively replaced. Costs are associated with each maintenance action, such as replacement after failure or preventive replacement. We derive a weak derivative estimator for the derivative of the cost performance with respect to θ. The technique is quite general and can be applied to many other threshold optimization problems in maintenance. The estimator is easy to implement and considerably increases the efficiency of a Robbins-Monro type of stochastic approximation algorithm. The paper is self-contained in the sense that it includes a proof of the correctness of the weak derivative estimation algorithm.

For a random variable (RV) X its moment index κ(X) ≡ sup{κ : E(|X|κ) < ∞} lies in 0 ≤ κ (X) ≤∞; it is a critical quantity and finite for heavy-tailed RVs. The paper shows that κ (min(X, Y)) ≥ κ(X) + κ (Y) for independent non-negative RVs X and Y. For independent non-negative ‘excess' RVs Xs and Ys whose distributions are the integrated tails of X and Y, κ (X) + κ (Y) ≤ κ (min(Xs, Ys)) + 2 ≤ κ (min(X, Y)). An example shows that the inequalities can be strict, though not if the tail of the distribution of either X or Y is a regularly varying function.

The stochastic logistic model has been studied in various contexts, including epidemiology, population biology, chemistry and sociology. Among the model predictions, the quasistationary distribution and the mean time to extinction are of major interest for most applications, and a number of approximation formulae for these quantities have been derived. In this paper, previous approximation formulae are improved for two mathematically tractable cases: at the limit of the number of individuals N → ∞ (with relative error of the approximations of the order 𝒪(1/N)), and at the limit of the basic reproduction ratio R0 → ∞ (with relative error of the approximations of the order 𝒪(1/R0)). The mathematically rigorous formulae are then extended heuristically for other values of N and R0 > 1.

Univariate probability inequalities have received extensive attention. It has been shown that under certain conditions, product-type bounds are valid and sharper than summation-type bounds. Although results concerning multivariate inequalities have appeared in the literature, product-type bounds in a multivariate setting have not yet been studied. This note explores an approach using graph theory and linear programming techniques to construct product-type lower bounds for the probability of the intersection among unions of k sets of events.

We consider a system where units having magnitudes arrive according to a nonhomogeneous Poisson process, remain there for a random period and then depart. Eventually, at any point in time only a portion of those units which have entered the system remain. Of interest are the finite time properties and limiting behaviors of the distribution of magnitudes among the units present in the system and among those which have departed from the system. We will derive limiting results for the empirical distribution of magnitudes among the active (departed) units. These results are also shown to extend to systems having stages or steps through which units must proceed. Examples are given to illustrate these results.

Taking up a recent proposal by Stadje and Parthasarathy in the setting of the many-server Poisson queue, we consider the integral ∫0∞[limu→∞E(X(u))-E(X(t))]dt as a measure of the speed of convergence towards stationarity of the process {X(t), t≥0}, and evaluate the integral explicitly in terms of the parameters of the process in the case that {X(t), t≥0} is an ergodic birth-death process on {0,1,….} starting in 0. We also discuss the discrete-time counterpart of this result, and examine some specific examples.

The paper proves the statement of the title, and shows that it has useful applications in evaluating the convergence of queueing models and Gibbs samplers with deterministic and random scans.

Let be the scan statistic of window size r for a sequence of n bistate trials . The scan statistic Sn(r) has been successfully used in various fields of applied probability and statistics, and its distribution has been studied extensively in the literature. Currently, all existing formulae for the distribution of Sn(r) are rather complex, and they can only be numerically implemented when is a sequence of Bernoulli trials, the window size r is less than 20 and the length of the sequence n is not too large. Hence, these formulae have been limiting the practical applications of the scan statistic. In this article, we derive a simple and effective formula for the distribution of Sn(r) via the finite Markov chain embedding technique to overcome some of the limitations of the existing complex formulae. This new formula can be applied when is either a sequence of Bernoulli trials or a sequence of Markov dependent bistate trials. Selected numerical examples are given to illustrate our results.