A method is provided for numerical evaluation, with any given accuracy, of the probability that at least p% of the genetic material from an individual's chromosomal segment survives to the next generation. Relevant MAPLE® V codes, for automated implementation of such evaluation, are also provided. The genomic continuum model, with Haldane's model for the crossover process, is assumed.

We derive an integral expression for the expected coverage of an n-dimensional target when the value density function of the target, the damage function, and the distribution of the damage function center around the target center are spherically symmetric. This integral expression is transformed by means of Parseval's theorem for Hankel transforms to an integral of the product of the Hankel transforms of order ½n − 1 of all underlying functions.

We consider integrals on Wiener space of the forms E(exp K(x)) and E(exp K(x) |L(x) = l) where K is a quadratic form and L a system of linear forms. We give explicit formulas for these integrals in terms of the operators K and L, in the case that these arise from quasilinear functions in the sense of Zhao (1981). As examples, we recover Lévy's area formula in the plane, and derive new formulas for the probability density of the radius of gyration tensor for Brownian paths.

For indivisible strong work-conserving queueing models with a Poisson arrival process, each of the following two properties is a sufficient condition for insensitivity. (1) The completed workload of a job receiving service is independent of the number of jobs in the system. (2) Independently of the completed workloads of the jobs in the system, they all are equally likely to be in service. For models which additionally belong to the class described by two families of parameters each of these properties is also necessary for insensitivity.

It is well known that a simple closed-form formula exists for the stationary distribution of the workload in M/GI/1 queues. In this paper, we extend this to the general stationary framework. Namely, we consider a work-conserving single-server queueing system, where the sequence of customers’ arrival epochs and their service times is described as a general stationary marked point process, and we derive a closed-form formula for the stationary workload distribution. The key to our proof is two-fold: one is the Palm-martingale calculus, that is, the connection between the notion of Palm probability and that of stochastic intensity. The other is the preemptive-resume last-come, first-served discipline.

A filtering problem for a branching process is studied within the tree framework, and the minimal intensity of the splitting process is given.

In this paper we show that to each distance d defined on the finite state space S of a strongly ergodic Markov chain there corresponds a coefficient ρd of ergodicity based on the Wasserstein metric. For a class of stochastically monotone transition matrices P, the infimum over all such coefficients is given by the spectral radius of P – R, where R = limkPk and is attained. This result has a probabilistic interpretation of a control of the speed of convergence of by the metric d and is linked to the second eigenvalue of P.

In this paper we study players' long-run behaviors in evolutionary coordination games with imperfect monitoring. In each time period, signals corresponding to players' underlying actions, instead of the actions themselves, are observed. A boundedly rational quasi-Bayesian learning process is proposed to extract information from the realized signals. We find that players' long-run behaviors depend not only on the correlations between actions and signals, but on the initial probabilities of risk-dominant and non-risk-dominant equilibria being chosen. The conditions under which risk-dominant equilibrium, non-risk-dominant equilibrium, and the coexistence of both equilibria emerges in the long run are shown. In some situations, the number of limiting distributions grows unboundedly as the population size grows to infinity.

We construct a wavelet-based almost-sure uniform approximation of fractional Brownian motion (FBM) (Bt(H))_t∈[0,1] of Hurst index H ∈ (0, 1). Our results show that, by Haar wavelets which merely have one vanishing moment, an almost-sure uniform expansion of FBM for H ∈ (0, 1) can be established. The convergence rate of our approximation is derived. We also describe a parallel algorithm that generates sample paths of an FBM efficiently.

We study the asymptotic tail behavior of the first-passage time over a moving boundary for a random walk conditioned to return to zero, where the increments of the random walk have finite variance. Typically, the asymptotic tail behavior may be described through a regularly varying function with exponent -½, where the impact of the boundary is captured by the slowly varying function. Yet, the moving boundary may have a stronger effect when the tail is considered at a time close to the return point of the random walk bridge, leading to a possible phase transition depending on the order of the distance between zero and the moving boundary.

It has been known for nearly a decade that deterministically modeled reaction networks that are weakly reversible and consist of a single linkage class have trajectories that are bounded from both above and below by positive constants (so long as the initial condition has strictly positive components). It is conjectured that the stochastically modeled analogs of these systems are positive recurrent. We prove this conjecture in the affirmative under the following additional assumptions: (i) the system is binary, and (ii) for each species, there is a complex (vertex in the associated reaction diagram) that is a multiple of that species. To show this result, a new proof technique is developed in which we study the recurrence properties of the n-step embedded discrete-time Markov chain.

The two-dimensional periodogram has been proposed as an estimator of the spectral density of a real, homogeneous, random field defined over a regular lattice on the plane. In the present paper, results pertaining to the asymptotic distributional properties of such a periodogram are obtained. These results generalize some of the work of Hannan (1960), Walker (1965) and more directly Olshen (1967a,b) concerning asymptotic theory for the periodogram of a stationary time series. Although extension of asymptotic theory for one-dimensional periodograms to a parallel theory for two-dimensional periodograms is not completely straightforward (one runs into problems akin to the problems encountered in extending the theory of one-dimensional trigonometric series to two dimensions), further extensions to asymptotic theory for p-dimensional periodograms (p > 2) is easily accomplished by an obvious mimicry of the definitions, theorems and proofs for the two-dimensional case. Since the notation required for the p-dimensional case is rather cumbersome, we have chosen to give results only for two-dimensional periodograms.

We consider a multicommodity flow problem on a complete graph whose edges have random, independent, and identically distributed capacities. We show that, as the number of nodes tends to infinity, the maximum utility, given by the average of a concave function of each commodity flow, has an almost-sure limit. Furthermore, the asymptotically optimal flow uses only direct and two-hop paths, and can be obtained in a distributed manner.

We consider a Lévy process Y(t) that is not continuously observed, but rather inspected at Poisson( $\omega$ ) moments only, over an exponentially distributed time $T_\beta$ with parameter $\beta$ . The focus lies on the analysis of the distribution of the running maximum at such inspection moments up to $T_\beta$ , denoted by $Y_{\beta,\omega}$ . Our main result is a decomposition: we derive a remarkable distributional equality that contains $Y_{\beta,\omega}$ as well as the running maximum process $\bar Y(t)$ at the exponentially distributed times $T_\beta$ and $T_{\beta+\omega}$ . Concretely, $\overline{Y}(T_\beta)$ can be written as the sum of two independent random variables that are distributed as $Y_{\beta,\omega}$ and $\overline{Y}(T_{\beta+\omega})$ . The distribution of $Y_{\beta,\omega}$ can be identified more explicitly in the two special cases of a spectrally positive and a spectrally negative Lévy process. As an illustrative example of the potential of our results, we show how to determine the asymptotic behavior of the bankruptcy probability in the Cramér–Lundberg insurance risk model.

This paper builds a mixture representation of the reliability function of the conditional residual lifetime of a coherent system in terms of the reliability functions of conditional residual lifetimes of order statistics. Some stochastic ordering properties for the conditional residual lifetime of a coherent system with independent and identically distributed components are obtained, based on the stochastically ordered coefficient vectors.

Consider an absolutely continuous distribution on [0, ∞) with finite mean μ and hazard rate function h(t) ≤ b for all t. For bμ close to 1, we would expect F to be approximately exponential. In this paper we obtain sharp bounds for the Kolmogorov distance between F and an exponential distribution with mean μ, as well as between F and an exponential distribution with failure rate b. We apply these bounds to several examples. Applications are presented to geometric convolutions, birth and death processes, first-passage times, and to decreasing mean residual life distributions.

Identifiability of evolutionary tree models has been a recent topic of discussion and some models have been shown to be nonidentifiable. A coalescent-based rooted population tree model, originally proposed by Nielsen et al. (1998), has been used by many authors in the last few years and is a simple tool to accurately model the changes in allele frequencies in the tree. However, the identifiability of this model has never been proven. Here we prove this model to be identifiable by showing that the model parameters can be expressed as functions of the probability distributions of subsamples, assuming that there are at least two (haploid) individuals sampled from each population. This a step toward proving the consistency of the maximum likelihood estimator of the population tree based on this model.

A new approach to the minimization of polyhedral convex functions is applied to give a finite algorithm for the rank regression problem. Numerical results for the Daniel and Wood example are presented.

The reversed strict ascending ladder epoch v1 + ··· +vm of the random walk S(n) with drift to ∞ is the m th time n at which the event S(n) < S(k), k > n, occurs and the corresponding ladder height is H1 + ··· + Hm = S(v1 + ··· + vm). It is shown that the random vectors (vi, Hi) are independent and for i ≧ 2 have the same distribution as the first strict ascending ladder time and height in the usual sense. This leads to an equality between the distribution of the first strict ladder height and P{min S(n) ≧ x} for x > 0. Reversed weak ladder points are defined analogously.

The negative exponential distribution is characterized in terms of two independent random variables. Only one of the random variables has a negative exponential distribution whilst the other can belong to a wide class of distributions. This result is then applied to two models for the reliability of a system of two modules subject to revealed and unrevealed faults to show when the models are equivalent. It is also shown, under certain conditions, that the system availability is only independent of the distribution of revealed failure times in one module when unrevealed failure times in the other module have a negative exponential distribution.