The limit distribution of the k th maximum from a random sample of size n when n → ∞ is identified as the distribution of the k th lower record value from one of three extreme value distributions. This fact is used in giving a different canonical representation and new proofs of the results of Hall (1978) for this limiting random variable. A characterization of the exponential distribution based on upper record values is given.

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.

A non-linear stationary stochastic process {Xt} is derived and shown to have the property that both the processes {Xt} and {log Xt} have the same correlation structure, viz. the Markov or first-order autoregressive correlation structure. The generation of such processes is discussed briefly and a characterization of the gamma distribution is obtained.

Moment recursion relations have previously been derived for the stationary probability density functions of continuous-time stochastic systems with Wiener (white noise) input. These results are extended in this paper to the case of Poisson (shot noise) input. The non-linear dynamical systems are expressed, in general, as stochastic differential equations, with an independent increment input. The transition probability density function evolves according to the appropriate Kolmogorov equation. Moments of the stationary density are obtained from the Fourier transform of the stationary density. The moment relations can be used to estimate the parameters of linear and non-linear stochastic systems from empirical moments, given either Wiener or Poisson input.

Generalized semi-Markov schemes were devised to give a versatile general model embracing queueing networks and similar systems of practical importance, and they have proved particularly successful in uniting many disparate results on insensitivity. However, it turns out that, although closed queueing networks are expressible as GSMS, open networks are not, and that the insensitivity results for such networks are not therefore strictly within their scope. In this paper, it is shown that, as one might hope, open networks can be realized as limits of a suitable sequence of closed networks in such a way that the insensitivity properties of the GSMS are transferred to the open network in the limit, and thus that open networks too can, in a sense, be considered to be GSMS. However, it appears from the technical nature of the arguments involved that, despite this close relationship between GSMS and open networks, it may nonetheless be simpler to treat them separately when constructing the proofs of theorems.

The present value is studied when I(t) is a stationary random income stream. The stationary distribution of V(t) for a family of simple streams modeled by stationary finite Markov chains is given explicitly. The process V(t) is shown to be observable in a special sense when I(t) is time-reversible.

There are m urns each containing k cells; n balls are assigned to the m urns in such a way that each cell contains at most one ball. Let Mt be the number of urns containing exactly t balls (t = 0, 1, •••, k). In the paper, the distribution of Mt, its moments and moment-generating function are obtained. The derivation of the last two seems to be new.

Null-recurrent random walks with symmetric step distribution are mapped to give null-recurrent Markov chains on the interval (0, 1). These chains are used to construct stationary Wold processes (point processes with Markov-dependent intervals) with infinite intensity, which processes are simple (i.e., almost surely orderly) but not (analytically) orderly. As examples of point processes with infinite intensity, they are constructed without any intermediate step that relies on using point processes of finite intensity.

Using Baddeley's [1] extension of Crofton's differential equation we derive an elementary integral formula for the probability that three randomly chosen points in the unit n-ball in ℝn, with respect to Lebesgue measure, form an acute triangle. When the dimension is 2 this probability is 4/π2 − 1/8, while when the dimension is 3 it is 33/70.

A direct proof of the expression for the limit probability generating function (p.g.f.) of the sum of Markov Bernoulli random variables is outlined. This depends on the larger eigenvalue of the transition probability matrix of their Markov chain.

Ross (1978) conjectured that the average delay in a single-server queue is larger when the arrival process is a non-stationary Poisson process than when it is a stationary Poisson process with the same rate. We present an example where equality obtains. When the number of waiting-positions is finite, Ross conjectured that the proportion of lost customers is greater in the nonstationary case. We present a counterexample to this conjecture.

Families of alternative bandit processes have been used as models for problems in a variety of areas. Optimal strategies for these decision processes are determined by dynamic allocation indices. These indices are here shown to play an important role in the evaluation of suboptimal strategies.

This paper discusses the asymptotic behaviour, and in particular the critical singularities, of the generating functions for Pólya walks with a given fraction of visits to a hyperplane when the walk is or is not constrained to be on one side of that hyperplane. The three-dimensional case has applications in physics and chemistry.