The Laplace transform of the extinction time is determined for a general birth and death process with arbitrary catastrophe rate and catastrophe size distribution. It is assumed only that the birth rates satisfy λ0 = 0, λ j > 0 for each j > 0, and . Necessary and sufficient conditions for certain extinction of the population are derived. The results are applied to the linear birth and death process (λ j = jλ, µj = jμ) with catastrophes of several different types.

A simple proof is given of the result that the ‘overflow' from a finite-state birth and death process is a renewal stream characterized by hyperexponential inter-event times. Our structure is utilized to give a converse result that any hyperexponential renewal stream can be so produced as the overflow from a finite-state birth and death process.

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.

Let {Yn: n ≧ 1} be a Harris-recurrent Markov chain on a general state space. It is shown that {Yn} is strong mixing, provided there exists a stationary probability distribution π (·) for {Yn}. Necessary and sufficient conditions for an autoregressive process to be uniform mixing are given.

Brownian motion subject to a quadratic killing rate and its connection with the Weibull distribution is analyzed. The distribution obtained for the process killing time significantly generalizes the Weibull. The derivation involves the use of the Karhunen–Loève expansion for Brownian motion, special function theory, and the calculus of residues.

The best upper and lower bounds for any probability generating function with mean m and finite variance are derived within the family of fractional linear functions with mean m. These are often intractable and simpler bounds, more useful for practical purposes, are then constructed. Direct applications in branching and epidemic theories are briefly presented; a slight improvement of the bounds is obtained for infinitely divisible distributions.

The Lewis-Thompson (L-T) approach to ordering distributions in terms of dispersion is compared to the dispersion ordering obtained from majorisation. For given conditions, L-T ordering is shown to be stronger than majorisation ordering. Dilation is considered in relation to majorisation ordering.

The problem of estimating the exponent of a stable law is receiving an increasing amount of attention because Pareto's law (or Zipf's law) describes many biological phenomena very well (see e.g. Hill (1974)). This problem was first solved by Hill (1975), who proposed an estimate, and the convergence of that estimate to some positive and finite number was shown to be a characteristic of distribution functions belonging to the Fréchet domain of attraction (Mason (1982)). As a contribution to a complete theory of inference for the upper tail of a general distribution function, we give the asymptotic behavior (weak and strong) of Hill's estimate when the associated distribution function belongs to the Gumbel domain of attraction. Examples, applications and simulations are given.

We extend some results of the extreme-value theory of stationary random sequences to non-stationary random sequences. The extremal index, defined in the stationary case, plays a similar role in the extended case. The details show that this index describes not only the behaviour of exceedances above a high level but also above a non-constant high boundary.

This paper describes the computer simulation of a coverage problem in geometric probability, that of placing random caps on the surface of a sphere. The simulation results were compared with exact values, where known, and the differences were negligible. This suggested the use of simulation results to assess several approximation formulas in the literature.

We find a ‘natural' measure on the set of convex infinite cylinders touching a convex body K in En. Using the curvature measures, we can calculate the probability that a convex infinite cylinder and a convex body K are tangent in some previously determined regions. As a particular case we obtain a measure on all q-planes touching K.

This paper investigates a system whose deterioration is expressed as a continuous-time Markov process. It is assumed that the state of the system cannot be identified without inspection. This paper derives an optimal policy minimizing the expected total long-run average cost per unit time. It gives the optimal time interval between successive inspections and determines the states at which the system is to be replaced. Furthermore, under some reasonable assumptions reflecting the practical meaning of the deterioration, it is shown that the optimal policy has monotonic properties. A control limit rule holds for replacement, and the time interval between successive inspections decreases as the degree of deterioration increases.

Objects of various integer sizes, o1, · ··, on, are to be packed together into bins of size N as they arrive at a service facility. The number of objects of size oi which arrive by time t is , where the components of are independent renewal processes, with At/t → λ as t → ∞. The empty space in those bins which are neither empty nor full at time t is called the wasted space and the system is declared stabilizable if for some finite B there exists a bin-packing algorithm whose use guarantees the expected wasted space is less than B for all t. We show that the system is stabilizable if the arrival processes are Poisson and λ lies in the interior of a certain convex polyhedral cone Λ. In this case there exists a bin-packing algorithm which stabilizes the system without needing to know λ. However, if λ lies on the boundary of Λ the wasted space grows as and if λ is exterior to Λ it grows as O(t); these conclusions hold even if objects may be repacked as often as desired.

This note deals with a purely algebraic development of the Spitzer identity for a bivariate Markov process , which is important for random walks, dams and queues. This identity, comprising the results of Kingman on the process , has been known for a long time in an analytic form as a solution of a certain integral equation. The algebraic approach leads to explicit results of interesting structure. In particular, the bivariate distribution of (Yn, Zn) can be expressed in terms of the marginal distributions of the stochastically dependent variables Yn and Zn.

A characteristic property of the Ewen sampling formula is shown to follow from the invariance under size-biased sampling of the Poisson–Dirichlet distribution.

Sufficient conditions are presented for the limiting normality of sequences of discrete random variables possessing unimodal distributions. The conditions are applied to obtain normal approximations directly for the hypergeometric distribution and the stationary distribution of a special birth-death process.

The limit of a product of independent 2 × 2 stochastic matrices is given when the entries of the first column are independent and have the same symmetric beta distribution. The rate of convergence is considered by introducing a stopping time for which asymptotics are given.

We show that the periodogram behaves differently for a weakly dependent process with a small monotonic trend and a stationary strongly dependent process. In the former case it has a non-central -distribution with noncentrality parameter tending to 0 uniformly outside intervals . In the latter case it has λα −1 times a central -distribution, 0 < α < 1.

Recently, unbiased stereological estimators of moments of particle volume, based on measurements on lower-dimensional sections through the particles, have been developed by Jensen and Gundersen (1985). In this note, we derive an explicit form of these unbiased estimators valid for particles in Rn of ellipsoidal shape, and we establish a close relationship between the estimators and a known integral geometric formula for ellipsoids, due to Furstenberg and Tzkoni (1971). Furthermore, a stereological version of another integral geometric formula for n-dimensional ellipsoids, due to Guggenheimer (1973), is derived.

A characterization of the exponential distribution in the class of all distributions which are HNBUE or HNWUE is proved. An upper (a lower) bound is obtained on the renewal function of a renewal process when the underlying distribution is HNBUE (HNWUE).