We establish some interesting duality results for Markov-modulated fluid flow models. Though fluid flow models are continuous-state analogues of quasi-birth-and-death processes, some duality results do differ by the inclusion of a scaling factor.

In this paper we first prove an arrangement-decreasing property of partial sums of independent random variables when they are partially ordered through the likelihood ratio ordering. We then apply a similar argument to obtain a stochastic ordering of random processes via a comparison of their parameter functions, with special applications to Poisson and Wiener processes. Finally, in Section 4 we present some applications in reliability theory, queueing, and first-passage problems.

Limit Statements obtainable by the key renewal theorem are of the form EXt = v(t) + o(1), as t →∞. We show how to delineate the limit function v for processes X associated with crudely regenerative phenomena. Included are refinements of classical limit theorems for Markov and regenerative processes, limits of sums of stationary random variables, and limits for integrals and derivatives of EXt.

Let {Zn} be a Galton-Watson process with i.i.d. random environments. This paper is concerned with estimation of π= E log Mn, where Mn is the. conditional expected number of offspring per nth generation individual, given the environments, when this quantity is positive. We show that, given non-extinction, {n-1 log Zn} is asymptotically the most efficient estimator of πamongst a broad class of those that are linear combinations of {log Zi: 1 ≦ j ≦ n}·

We look forwards and backwards in the multi-allelic neutral exchangeable Cannings model with fixed population size and nonoverlapping generations. The Markov chain X is studied which describes the allelic composition of the population forward in time. A duality relation (inversion formula) between the transition matrix of X and an appropriate backward matrix is discussed. The probabilities of the backward matrix are explicitly expressed in terms of the offspring distribution, complementing the work of Gladstien (1978). The results are applied to fundamental multi-allelic Cannings models, among them the Moran model, the Wright-Fisher model, the Kimura model, and the Karlin and McGregor model. As a side effect, number theoretical sieve formulae occur in these examples.

In a decision problem with uncertainty a decision maker receives partial information about the actual state via an information structure. After receiving a signal, he is allowed to withdraw and gets zero profit. We say that one structure is better than another when a withdrawal option exists if it may never happen that one structure guarantees a positive profit while the other structure guarantees only zero profit. This order between information structures is characterized in terms that are different from those used by Blackwell's comparison of experiments. We also treat the case of a malevolent nature that chooses a state in an adverse manner. It turns out that Blackwell's classical characterization also holds in this case.

Brown (1980), (1981) proved that the renewal function is concave if the interrenewal distribution is DFR (decreasing failure rate), and conjectured the converse. This note settles Brown's conjecture with a class of counterexamples. We also give a short proof of Shanthikumar's (1988) result that the DFR property is closed under geometric compounding.

For a bivariate Lévy process (ξt,ηt)t≥ 0 and initial value V0 define the generalised Ornstein–Uhlenbeck (GOU) process Vt:=eξt (V0+∫t0 e-ξs-dηs), t≥0, and the associated stochastic integral process Zt:=∫0t e-ξs-dηs, t≥0. Let Tz:=inf{t>0: Vt<0|V0=z} and ψ(z):=P(Tz<∞) for z≥0 be the ruin time and infinite horizon ruin probability of the GOU process. Our results extend previous work of Nyrhinen (2001) and others to give asymptotic estimates for ψ(z) and the distribution of Tz as z→∞, under very general, easily checkable, assumptions, when ξ satisfies a Cramér condition.

We observe that the technique of Markov contraction can be used to establish measure concentration for a broad class of noncontracting chains. In particular, geometric ergodicity provides a simple and versatile framework. This leads to a short, elementary proof of a general concentration inequality for Markov and hidden Markov chains, which supersedes some of the known results and easily extends to other processes such as Markov trees. As applications, we provide a Dvoretzky-Kiefer-Wolfowitz-type inequality and a uniform Chernoff bound. All of our bounds are dimension-free and hold for countably infinite state spaces.

In this paper a method based on a Markov chain Monte Carlo (MCMC) algorithm is proposed to compute the probability of a rare event. The conditional distribution of the underlying process given that the rare event occurs has the probability of the rare event as its normalizing constant. Using the MCMC methodology, a Markov chain is simulated, with the aforementioned conditional distribution as its invariant distribution, and information about the normalizing constant is extracted from its trajectory. The algorithm is described in full generality and applied to the problem of computing the probability that a heavy-tailed random walk exceeds a high threshold. An unbiased estimator of the reciprocal probability is constructed whose normalized variance vanishes asymptotically. The algorithm is extended to random sums and its performance is illustrated numerically and compared to existing importance sampling algorithms.

A monotone coupling of order statistics from two sets of independent non-negative random variables Xi, i = 1, ···, n, and Yi, i = 1, ···, n, means that there exist random variables X′i, Y′i, i = 1, ···, n, on a common probability space such that , and a.s. j = 1, ···, n, where X(1) ≦ X(2) ≦ ·· ·≦ X(n) are the order statistics of Xi, i = 1, ···, n (with the corresponding notations for the X′, Y, Y′ sample). In this paper, we study the monotone coupling of order statistics of lifetimes in two multi-unit systems under multivariate imperfect repair. Similar results for a special model due to Ross are also given.

Let $X_1,X_2, \ldots, X_n$ be a sequence of independent random points in $\mathbb{R}^d$ with common Lebesgue density f. Under some conditions on f, we obtain a Poisson limit theorem, as $n \to \infty$ , for the number of large probability kth-nearest neighbor balls of $X_1,\ldots, X_n$ . Our result generalizes Theorem 2.2 of [11], which refers to the special case $k=1$ . Our proof is completely different since it employs the Chen–Stein method instead of the method of moments. Moreover, we obtain a rate of convergence for the Poisson approximation.

The computer age and the phenomenological complexity of the AIDS/HIV epidemic have engendered a rich profusion of deterministic and stochastic time series models for the development of an epidemic. The present study examines the reliability of deterministic approximations of fundamentally random processes. Through numerical analysis and probabilistic considerations, we derive absolute and simultaneous confidence interval bounding techniques, and offer a practical procedure based on these developments. A heartening aspect of the computational study presented at the close of this paper indicates that when the population size is in the thousands, the deterministic version to the classical logistic epidemic is a good approximation.

The theory of weak convergence has developed into an extensive and useful, but technical, subject. One of its most important applications is in the study of empirical distribution functions: the explication of the asymptotic behavior of the Kolmogorov goodness-of-fit statistic is one of its greatest successes. In this article a simple method for understanding this aspect of the subject is sketched. The starting point is Doob's heuristic approach to the Kolmogorov-Smirnov theorems, and the rigorous justification of that approach offered by Donsker. The ideas can be carried over to other applications of weak convergence theory.

A coherent system is observed until it fails. At the instant of system failure, the set of failed components and the failure time of the system are noted. The failure times of the components are not known. We consider whether the component life distributions can be determined from the distributions of the observed data.

Meilijson (1981) gave a condition on the structure of the system that was sufficient for the identifiability of the component distributions, under the assumption that the component life distributions are continuous and have common essential extrema. Nowik (1990) gave necessary and sufficient conditions for identifiability under the more restrictive condition that the component distributions have atoms at their common essential infimum and are mutually absolutely continuous. We give a necessary condition for identifiability, which we show to be equivalent to Nowik's condition, under the assumption that the distributions are continuous and strictly increasing. We derive a sufficient condition for identifiability, more general than Meilijson's, for the case in which the component distributions are assumed to be analytic. We also show that our necessary condition for identifiability is both necessary and sufficient when the component life distributions are assumed to belong to certain parametric families.

A multiclass c-server retrial queueing system in which customers arrive according to a class-dependent Markovian arrival process (MAP) is considered. Service and retrial times follow class-dependent phase-type (PH) distributions with the further assumption that PH distributions of retrial times are acyclic. A necessary and sufficient condition for ergodicity is obtained from criteria based on drifts. The infinite state space of the model is truncated with an appropriately chosen Lyapunov function. The truncated model is described as a multidimensional Markov chain, and a Kronecker representation of its generator matrix is numerically analyzed.

This paper considers the generalized likelihood ratio (GLR) test or its modification dealing with nested models. The algorithm for evaluating the critical values and the error-rates for the canonical tests are provided; a table of critical values of a class of GLR tests is also given. The test proposed in the paper has applications in time-series model selection.

“Recursive” games were first defined and studied by Everett. Related results can be found in Gillette, Milnor and Shapley, and Blackwell and Ferguson. In this paper we introduce the notion of a recursive matrix game, which we believe eliminates the vagueness but none of the useful generality of the earlier definition. We then give an inductive proof (different from the proof in [3]) that these games have a value, with ∊-optimal stationary strategies available to each player. We also apply the result and show how a class of games studied in a different framework are games of this type and thus have a value.

The general minimum mean square error prediction problem is studied for the class of processes defined by polynomial functions of the Ornstein-Uhlenbeck process. In particular, the relative error difference for non-linear prediction compared with optimal linear prediction is calculated. Limiting values are determined, including an upper bound for the maximal relative error difference. These results show no difference over the entire class for both long and short prediction lead times. For a single Hermite polynomial, the relative error difference is zero. An analytical solution for the maximal difference for lead times below a given threshold is derived. This latter result has been verified numerically on a digital computer for polynomials up to 17th degree.

This paper continues the author's study of age-dependent branching processes allowing immigration. In this paper the multidimensional case is considered. A sufficient condition is obtained for the existence of a legitimate limiting distribution. Several corollaries are obtained, which generalize many of the results of the discrete theory and those of the one-dimensional continuous time model.