An alternative version of the necessary and sufficient condition for almost sure fixation in the conditional branching process model is derived. This formulation provides an insight into why the examples considered in Buckley and Seneta (1983) all have the same condition for fixation.

A new and simpler proof of Morrison's result that within exponential mixtures only IFR gamma mixing produces linearly increasing mean residual life functions is given. A parallel and new characterization of the DFR gamma laws follows as a consequence. The method of proof used suggests a general result on the infinite divisibility of the mixing distributions in exponential mixtures.

The mean square and almost sure convergence of W(t) = e–αtZ(t) is proved for a super-critical multitype age-dependent branching process allowing immigration at the epochs of a renewal process. It is shown that the Malthusian parameter, asymptotic frequencies of types and stationary age distributions are the same for the processes with and without immigration.

Let {Zn} be a finite mean supercritical Bienaymé– Galton–Watson process. It is known that there exist norming constants {Cn} such that {Zn/Cn} converges almost surely to a limit W. Also there is a whole literature concerning properties of {Cn} and W. We attempt a new approach to the limit theory of {Zn} by relating it to the theory of sums of independent and identically distributed random variables.

Processes whose entities would be independent but whose behaviour is influenced by environmental variables, which they in turn affect, are processes with mediated interactions. The independent open migration process with which is associated a random variable, the excitation, is such a process. This self-excited (conditionally) independent open migration process, with finite excitation, is related to a derived process, its piecewise-deterministic analogue. The correspondence gives insight into the process, as well as an equilibrium independence result. A simple example illustrates the presence of paroxysmal phenomena.

An identity, of the type of Green's equation, is deduced for the loss function of a stopping process. This yields a set of relations determining the optimal (minimal loss) stopping boundary, which do not require simultaneous determination of the loss function in the stopping region. A no-overshoot approximation is invoked, but a bound on the magnitude of the terms neglected is obtained by appeal to a general version of Chernoff's tangency condition at an optimal boundary.

In this work the $\ell_q$ -norms of points chosen uniformly at random in a centered regular simplex in high dimensions are studied. Berry–Esseen bounds in the regime $1\leq q < \infty$ are derived and complemented by a non-central limit theorem together with moderate and large deviations in the case where $q=\infty$ . An application to the intersection volume of a regular simplex with an $\ell_p^n$ -ball is also carried out.

Let X1, X2, · ··, Xn be independent random variables such that ai ≦ Xi ≦ bi, i = 1,2,…n. A class of upper bounds on the probability P(S−ES ≧ nδ) is derived where S = Σf(Xi), δ > 0 and f is a continuous convex function. Conditions for the exponential convergence of the bounds are discussed.

A stochastic theory is given for speculative industrial chemical research. It is shown how the theory may be used to suggest a suitable balance between long-term and short-term projects.

Kendall [2] gave a thorough discussion of De Finetti constants for a sequence (finite or infinite) of exchangeable events. Galambos [1] and Ridler-Rowe [3] also found some interesting results in this area. In this paper, we intend to give a necessary and sufficient condition for a finite sequence of exchangeable events to be extendable to an infinite sequence of exchangeable events.

It is well known (Lee 1960), especially to communications and control systems engineers, that the weighting function (unit impulse function) of a linear causal system may be obtained by cross correlating its output with its input. In general a Wiener-Hopf integral equation must be solved for the weighting function; but if the input were “white noise” of unit spectral density then, as shown by Lee (1950), the weighting function would be equal to this cross correlation function for positive values of its argument. Although a perfectly “white” input cannot be obtained in practice, a number of modifications of this technique have been used; Anderson, Buland and Cooper (1959), for example, describe the use of specially selected samples of “discrete-interval binary noise” in a problem in adaptive control.

The coupling method of Ball (1986) is extended to carrier-borne epidemics, thus providing a new proof of a result of Daniels (1972) concerning the total size distribution of Downton's stochastic carrier-borne epidemic. The generalization to multipopulation carrier-borne epidemics is immediate.

We present a brief summary of some results related to deriving orthogonal representations of second-order random fields and its application in solving linear prediction problems. In the homogeneous and/or isotropic case, the spectral theory provides an orthogonal expansion in terms of spherical harmonics, called spectral decomposition (Yadrenko 1983). A prediction formula based on this orthogonal representation is shown. Finally, an application of this formula in solving a real-data problem related to prospective geophysics techniques is presented.

The critical paths of a max-plus linear system with noise are random variables. In this paper we introduce the edge criticalities which measure how often the critical paths traverse each edge in the precedence graph. We also present the parallel path approximation, a novel method for approximating these new statistics as well as the previously studied max-plus exponent. We show that, for low amplitude noise, the critical paths spend most of their time traversing the deterministic maximally weighted cycle and that, as the noise amplitude is increased, the critical paths become more random and their distribution over the edges in the precedence graph approaches a highly uniform measure of maximal entropy.

Let X(t) be a continuous time Markov process on the integers such that, if σ is a time at which X makes a jump, X(σ)– X(σ–) is distributed independently of X(σ–), and has finite mean μ and variance. Let q(j) denote the residence time parameter for the state j. If tn denotes the time of the nth jump and Xn ≡ X(tb), it is easy to deduce limit theorems for from those for sums of independent identically distributed random variables. In this paper, it is shown how, for μ > 0 and for suitable q(·), these theorems can be translated into limit theorems for X(t), by using the continuous mapping theorem.

We introduce extensions to an infinite-allele branching process that allows for multiple types to exist alongside labels. We consider a Markov branching process and general branching process under different assumptions, and show asymptotic results about the growth of the labels as well as the frequency spectrum. These results are motivated by two separate models. The Markov binary splitting results are motivated by a model of clonal evolution in cancer that considers the effect of both driver and passenger mutations on tumor growth. The general process has applications in viral reproduction and dynamics.

In a counting process considered at time t the focus is often on the length of the current interarrival time, whereas points in the past may be said to constitute information about the process. The paper introduces new concepts on how to quantify predictability of the future behavior of counting processes based on the past information and considers then situations in which the future points become more (or less) predictable. Various properties of our proposed concepts are studied and applications relevant to the reliability of repairable systems are given.

Define a random variable

We refer to Yn as a sustained maximum. Under the assumption that the Xi's are i.i.d. the exact distribution of Yn is obtained. Under certain conditions on the underlying distribution F we obtain weak limit results for the Yn as well. Also a combinatorial extreme-value problem is solved.