We study a variety of optimal investment problems for objectives related to attaining goals by a fixed terminal time. We start by finding the policy that maximizes the probability of reaching a given wealth level by a given fixed terminal time, for the case where an investor can allocate his wealth at any time between n + 1 investment opportunities: n risky stocks, as well as a risk-free asset that has a positive return. This generalizes results recently obtained by Kulldorff and Heath for the case of a single investment opportunity. We then use this to solve related problems for cases where the investor has an external source of income, and where the investor is interested solely in beating the return of a given stochastic benchmark, as is sometimes the case in institutional money management. One of the benchmarks we consider for this last problem is that of the return of the optimal growth policy, for which the resulting controlled process is a supermartingale. Nevertheless, we still find an optimal strategy. For the general case, we provide a thorough analysis of the optimal strategy, and obtain new insights into the behavior of the optimal policy. For one special case, namely that of a single stock with constant coefficients, the optimal policy is independent of the underlying drift. We explain this by exhibiting a correspondence between the probability maximizing results and the pricing and hedging of a particular derivative security, known as a digital or binary option. In fact, we show that for this case, the optimal policy to maximize the probability of reaching a given value of wealth by a predetermined time is equivalent to simply buying a European digital option with a particular strike price and payoff. A similar result holds for the general case, but with the stock replaced by a particular (index) portfolio, namely the optimal growth or log-optimal portfolio.

A parametric family of completely random measures, which includes gamma random measures, positive stable random measures as well as inverse Gaussian measures, is defined. In order to develop models for clustered point patterns with dependencies between points, the family is used in a shot-noise construction as intensity measures for Cox processes. The resulting Cox processes are of Poisson cluster process type and include Poisson processes and ordinary Neyman-Scott processes.

We show characteristics of the completely random measures, illustrated by simulations, and derive moment and mixing properties for the shot-noise random measures. Finally statistical inference for shot-noise Cox processes is considered and some results on nearest-neighbour Markov properties are given.

For sequences of i.i.d. random variables whose common tail 1 – F is regularly varying at infinity wtih an unknown index –α < 0, it is well known that the Hill estimator is consistent for α–1 and usually asymptotically normally distributed. However, because the Hill estimator is a function of k = k(n), the number of upper order statistics used and which is only subject to the conditions k →∞, k/n → 0, its use in practice is problematic since there are few reliable guidelines about how to choose k. The purpose of this paper is to make the use of the Hill estimator more reliable through an averaging technique which reduces the asymptotic variance. As a direct result the range in which the smoothed estimator varies as a function of k decreases and the successful use of the esimator is made less dependent on the choice of k. A tail empirical process approach is used to prove the weak convergence of a process closely related to the Hill estimator. The smoothed version of the Hill estimator is a functional of the tail empirical process.

Let s1, …, sn be generated governed by an r-state irreducible aperiodic Markov chain. The partial sum process is determined by a realization of states with s0 = α and the real-valued i.i.d. bounded variables Xαß associated with the transitions si = α, si+1 = β. Assume Χ αβ has negative stationary mean. The explicit limit distribution of the maximal segmental sum is derived. Computational methods with potential applications to the analysis of random Markov-dependent letter sequences (e.g. DNA and protein sequences) are presented.

If Bn is the time of the first birth in the nth generation in a supercritical irreducible multitype Crump–Mode process then when there are people in every generation Bn/n converges to a constant; if Dn is the time of the last birth in the nth generation then Dn/n also converges to a constant on the survival set. Analogous results hold for the extreme members of the nth generation in a branching random walk.

Leslie (1945) models the evolution in discrete time of a closed, single-sex population with discrete age groups by multiplying a vector describing the age structure by a matrix containing the birth and death rates. We suppose that successive matrices are chosen according to a Markov chain from a finite set of matrices. We find exactly the long-run rate of growth and expected age structure. We give two approximations to the variance in age structure and total population size. A numerical example illustrates the ergodic features of the model using Monte Carlo simulation, finds the invariant distribution of age structure from a linear integral equation, and calculates the moments derived here.

Let Φ = {Φ n} be an aperiodic, positive recurrent Markov chain on a general state space, π its invariant probability measure and f ≧ 1. We consider the rate of (uniform) convergence of Ex[g(Φ n)] to the stationary limit π (g) for |g| ≦ f: specifically, we find conditions under which

as n →∞, for suitable subgeometric rate functions r. We give sufficient conditions for this convergence to hold in terms of

(i) the existence of suitably regular sets, i.e. sets on which (f, r)-modulated hitting time moments are bounded, and

(ii) the existence of (f, r)-modulated drift conditions (Foster–Lyapunov conditions).

The results are illustrated for random walks and for more general state space models.

In a paper of the same title, Ewens has considered the problem of inferring whether the genotypic frequencies observed in a small sample are consistent with a particular population model, in which all types are selectively neutral. The present paper gives the general theory of such sampling schemes, when the sampling is from either a deterministic, or a stochastically varying, population.

The theory is illustrated on a population model which was introduced by Karlin and McGregor. Various results are proved for both the population, and its sample. It is shown that Ewens' sampling theory applies to this population model, and conjectures as to why different population models may have the same sampling theory are made.

This paper is mainly concerned with the asymptotic theory of maximum likelihood estimation for continuous-time stochastic processes. The role of martingale limit theory in this theory is developed. Some analogues of classical statistical concepts and quantities are also suggested. Various examples that illustrate parts of the theory are worked through, producing new results in some cases. The role of diffusion approximations in estimation is also explored.

A new time series model for exponential variables having first-order autoregressive structure is presented. Unlike the recently studied standard autoregressive model in exponential variables (ear(1)), runs of constantly scaled values are avoidable, and the two parameter structure allows some adjustment of directional effects in sample path behaviour. The model is further developed by the use of cross-coupling and antithetic ideas to allow negative dependency. Joint distributions and autocorrelations are investigated. A transformed version of the model has a uniform marginal distribution and its correlation and regression structures are also obtained. Estimation aspects of the models are briefly considered.

The paper reviews various aspects, mostly mathematical, concerning the output or departure process of a general queueing system G/G/s/N with general arrival process, mutually independent service times, s servers (1 ≦ s ≦ ∞), and waiting room of size N (0 ≦ N ≦ ∞), subject to the assumption of being in a stable stationary condition. Known explicit results for the distribution of the stationary inter-departure intervals {Dn} for both infinite and finite-server systems are given, with some discussion on the use of reversibility in Markovian systems. Some detailed results for certain modified single-server M/G/1 systems are also available. Most of the known second-order properties of {Dn} depend on knowing that the system has either Poisson arrivals or exponential service times. The related stationary point process for which {Dn} is the stationary sequence of the corresponding Palm–Khinchin distribution is introduced and some of its second-order properties described. The final topic discussed concerns identifiability, and questions of characterizations of queueing systems in terms of the output process being a renewal process, or uncorrelated, or infinitely divisible.

The distribution of the size of one connected component and the largest connected component of the excursion set is derived for stationary χ2, t and F fields, in the limit of high or low thresholds. This extends previous results for stationary Gaussian fields (Nosko 1969, Adler 1981) and for χ2 fields in one and two dimensions (Aronowich and Adler 1986, 1988). An application of this is to detect regional changes in positron emission tomography (PET) images of blood flow in human brain, using the size of the largest connected component of the excursion set as a test statistic.

A general method based on a limit theorem for generation of random numbers from infinitely divisible distributions with essentially given Lévy measure is studied. Some classes of infinitely divisible distributions that appear in a natural way in this context are paid particular attention.

Queueing systems with distinguished arrivals are described on the basis of Markov arrival processes with marked transitions. Customers are distinguished by their types of arrival. Usually, the queues observed by customers of different types are different, especially for queueing systems with bursty arrival processes. We study queueing systems from the points of view of customers of different types. A detailed analysis of the fundamental period, queue lengths and waiting times at the epochs of arrivals is given. The results obtained are the generalizations of the results of the MAP/G/1 queue.

Recently, there has been considerable interest in some specialized binary lattice processes. However, this exciting work has been rather fragmentary and heuristic. Rigorous proofs are provided for the existence of some classes of stationary unilateral Markov fields on infinite square lattices. (Neighbours are sites which are horizontally, vertically, or diagonally adjacent.) Many of the difficulties are avoided by characterizing stationary unilateral Markov fields on finite lattices first. Detailed analyses are given for both binary and Gaussian variables.

This paper shows how the Laplace transform analysis of Bailey (1954), (1957) can be continued to yield additional insights about the time-dependent behavior of the queue-length process in the M/M/1 model. A transform factorization is established that leads to a decomposition of the first moment as a function of time into two monotone components. This factorization facilitates developing approximations for the moments and determining their asymptotic behavior as . All descriptions of the transient behavior are expressed in terms of basic building blocks such as the first-passage-time distributions. The analysis is facilitated by appropriate scaling of space and time so that regulated or reflected Brownian motion (RBM) appears as the special case in which the traffic intensity ρ equals the critical value 1. An operational calculus is developed for obtaining M/M/1 results directly from corresponding RBM results as well as vice versa. The analysis thus provides useful insight about RBM approximations for queues.

A simple expression for the characteristic functional of generalized shot noise is developed. Through expansions in terms of functional derivatives this yields expressions for moment functions of all orders. A central limit theorem also follows. Several examples are discussed.

Replacement policies based on measurements of some increasing state variable, e.g. wear, accumulated damage or accumulated stress, are studied in this paper. It is assumed that the state measurements may be regarded as realizations of some stochastic process and that the proneness to failure of an active unit may be described by an increasing state-dependent failure rate function. Average long-run cost per unit time is considered. The optimal replacement rule is shown to be a control limit rule, i.e. it is optimal to replace either at failure or when the state variable has reached some threshold value, whichever occurs first. The optimal rule is determined. Some generalizations and special cases are given.

A revealing alternate proof is provided for the Iglehart (1965), (1973)–Borovkov (1967) heavy-traffic limit theorem for GI/G/s queues. This kind of heavy traffic is obtained by considering a sequence of GI/G/s systems with the numbers of servers and the arrival rates going to ∞ while the service-time distributions are held fixed. The theorem establishes convergence to a Gaussian process, which in general is not Markov, for an appropriate normalization of the sequence of stochastic processes representing the number of customers in service at arbitrary times. The key idea in the new proof is to consider service-time distributions that are randomly stopped sums of exponential phases, and then work with the discrete-time vector-valued Markov chain representing the number of customers in each phase of service at arrival epochs. It is then easy to show that this sequence of Markov chains converges to a multivariate O–U (Ornstein–Uhlenbeck) diffusion process by applying simple criteria in Stroock and Varadhan (1979). The Iglehart–Borovkov limit for these special service-time distributions is the sum of the components of this multivariate O–U process. Heavy-traffic convergence is also established for the steady-state distributions of GI/M/s queues under the same conditions by exploiting stochastic-order properties.

Shot noise Cox processes constitute a large class of Cox and Poisson cluster processes in ℝd, including Neyman-Scott, Poisson-gamma and shot noise G Cox processes. It is demonstrated that, due to the structure of such models, a number of useful and general results can easily be established. The focus is on the probabilistic aspects with a view to statistical applications, particularly results for summary statistics, reduced Palm distributions, simulation with or without edge effects, conditional simulation of the intensity function and local and spatial Markov properties.