We study Bose gases in $d \ge 2$ dimensions with short-range repulsive pair interactions at positive temperature, in the canonical ensemble and in the thermodynamic limit. We assume the presence of hard Poissonian obstacles and focus on the non-percolation regime. For sufficiently strong interparticle interactions, we show that almost surely there cannot be Bose–Einstein condensation into a sufficiently localized, normalized one-particle state. The results apply to the canonical eigenstates of the underlying one-particle Hamiltonian.

A new upper bound on the stationary waiting-time distribution of a GI/G/1 queue is derived following Kingman's martingale approach. This bound is generally stronger than Kingman's upper bound and is sometimes stronger than an upper bound derived by Ross.

We study a classical multitype Galton–Watson process with mutation and selection. The individuals are sequences of fixed length over a finite alphabet. On the sharp peak fitness landscape together with independent mutations per locus, we show that, as the length of the sequences goes to ∞ and the mutation probability goes to 0, the asymptotic relative frequency of the sequences differing on k digits from the master sequence approaches (σe-a - 1)(ak/k!)∑i≥ 1ik/σi, where σ is the selective advantage of the master sequence and a is the product of the length of the chains with the mutation probability. The probability distribution Q(σ, a) on the nonnegative integers given by the above equation is the quasispecies distribution with parameters σ and a.

This paper considers the two-sex birth-death model {X(t), Y(t); t ≧ 0}; an explicit solution is obtained for its probability generating function. It is shown that moments of the process can be found directly from the Kolmogorov forward equations for the probabilities. An integral equation approach is also used to throw light on the structure of the process.

Consider an oscillating integer valued random walk up to the first hitting time of some fixed integer x > 0. Suppose there is a fee to be paid each time the random walk crosses the level x, and that the amount corresponds to the overshoot. We determine the distribution of the sum of these fees in terms of the renewal functions of the ascending and descending ladder heights. The proof is based on the observation that some path transformation of the random walk enables us to translate the problem in terms of the intersection of certain regenerative sets.

The model Y(t) = s(t | θ) + ε(t) is studied in the case that observations are made at scattered points τ j in a subset of Rp and θ is a finite-dimensional parameter. The particular cases of 0 = (α, β) and (α, β, ω) are considered in detail. Consistency and asymptotic normality results are developed assuming that the spatial series ε(·) and the point process {τ j} are independent, stationary and mixing. The estimates considered are equivalent to least squares asymptotically and are not generally asymptotically efficient.

Contributions of the paper include: study of the Rp case, management of irregularly placed observations, allowance for abnormal domains of observation and the discovery that aliasing complications do not arise when the point process {τ j} is mixing. There is a brief discussion of the construction and properties of maximum likelihood estimates for the spatial-temporal case.

This note shows how the initial density of the volume of a random simplex in a body, K, depends upon the fourth moment of sectional area and so rejects the conjecture that this initial density is independent of K.

This paper is concerned with the area under the trajectory of carriers in the Downton carrier-borne epidemic (1968). Its Laplace–Stieltjes transform is studied, and a recursive equation derived for it. Its mean is examined in some detail and the solution obtained is expressed as a power series in π, the proportion of the infected susceptibles becoming carriers.

Hoeffding's inequality can be used in conjunction with the declared parameters of a traffic source, such as its peak rate, to obtain confidence intervals for measurements of the traffic's effective bandwidth. We describe a variety of interval-estimation procedures based on this idea, designed to provide differing degrees of robustness against non-stationarity. We also discuss how to compute confidence intervals for the effective bandwidth of an aggregate of traffic sources.

By making the time parameters of a birth and death process random variables, we create a continuous-time analogue of random walk in a random environment. Criteria for recurrence or transience are discussed and an a.s. convergence law is determined.

We consider a multiple regression model in which the regressors are Fourier cosine vectors. These regressors are intended as approximations to ‘slowly changing' regressors of the kind often found in time series regression applications. The errors in the model are assumed to be generated by a special type of autoregressive model defined so that the regressors are eigenvectors of the quadratic forms occurring in the exponent of the probability density of the errors. This autoregression is intended as an approximation to the usual stationary autoregression. Both approximations are adopted for the sake of mathematical convenience.

Student's t-statistics are constructed for the autoregressive coefficients in a manner analogous to ordinary regression. It is shown that these statistics are distributed as Student's t to the first order of approximation, that is with errors in the density of order T−1/2, where T is the sample size, while the squares of the statistics are distributed as the square of Student's t to the second order of approximation, that is with errors in the density of order Τ–1.

Dynamic stationary models for mixed time series and cross-section data are studied. The models are of simple, standard form except that the unknown coefficients are not assumed constant over the cross-section; instead, each cross-sectional unit draws a parameter set from an infinite population. The models are framed in continuous time, which facilitates the handling of irregularly-spaced series, and observation times that vary over the cross-section, and covers also standard cases in which observations at the same regularly-spaced times are available for each unit. A variety of issues are considered, in particular stationarity and distributional questions, inference about the parameter distributions, and the behaviour of cross-sectionally aggregated data.

Consider a finite or infinite collection of urns, each with capacity r, and balls randomly distributed among them. An overflow is the number of balls that are assigned to urns that already contain r balls. When $r=1$ , this is the number of balls landing in non-empty urns, which has been studied in the past. Our aim here is to use martingale methods to study the asymptotics of the overflow in the general situation, i.e. for arbitrary r. In particular, we provide sufficient conditions for both Poissonian and normal asymptotics.

We investigate the asymptotic behaviour of empirical processes truncated outside an interval about the (1 – s(n)/n)-quantile where s(n) → ∞ and s(n)/n → 0 as the sample size n tends to ∞. It is shown that extreme value (Poisson) processes and, alternatively, the homogeneous Poisson process may serve as approximations if certain von Mises conditions hold.

We consider a continuous Gaussian random field living on a compact set $T\subset \mathbb{R}^{d}$ . We are interested in designing an asymptotically efficient estimator of the probability that the integral of the exponential of the Gaussian process over T exceeds a large threshold u. We propose an Asmussen–Kroese conditional Monte Carlo type estimator and discuss its asymptotic properties according to the assumptions on the first and second moments of the Gaussian random field. We also provide a simulation study to illustrate its effectiveness and compare its performance with the importance sampling type estimator of Liu and Xu (2014a).

The motion of electrons through a gas of particles is modelled as a two-state process: an electron is alternatingly moving and being held captive by a non-moving particle, during exponentially distributed periods.

The model contains a parameter p, the probability that collision of an electron with a gas particle does not lead to its capture but produces an extra electron.

The quantity of interest is the current C(t) produced by the moving electrons, as a function of time.

We analyse several aspects of a class of simple counting processes that can emerge in some fields of applications where a change point occurs. In particular, under simple conditions we prove a significant inequality for the stochastic intensity.

Falin (1984) examined the quasi-input process (the flow of service starting times) in the M/G/1/∞ queue and raised the question as to whether this process is a renewal process. We show that, except in the trivial case of instantaneous service, the quasi-input process is never renewal.

It is known that if q vehicles at speed v pass a fixed point per unit time, randdomly and independently, and mix with another set q', v', then the expected number of overtakings per unit length of road per unit time is qq' (v' – v)/vv'. If k, k' are the concentrations, i.e., number per unit length of road, this rate can also be expressed as kk' (v'– v) and as q'k – qk'. It is also the expected number of intersections per unit area on the space-time diagram, where the vehicle paths are represented by sets of random parallel lines. If each set is represented by a vector with magnitude equal to the density (i.e., inverse of the spacing), the rate is equal to the magnitude of the vector product. This is extended to the general case of a stream of vehicles with distributed speeds. This has an equivalent vector, whose components are the total concentration (in the direction of the time axis) and total flow (distance direction). This leads to the concept of the cumulative vector curve in problems of geometrical probability. The extension to the case of variable flows and speeds is indicated. The vector representing the distribution of speeds is then variable, but satisfies a condition of continuity which makes its divergence vanish.

We generalize a theorem due to Keilson on the approximate exponentiality of waiting times for rare events in regenerative processes. We use the result to investigate the limit distribution for a family of first entrance times in a sequence of Ehrenfest urn models. As a second application, we consider approximate pattern matching, a problem arising in molecular biology and other areas.