This paper is motivated by the modelling of leaching of bacteria through soil. A semi-linear process X t − may be used to describe the soil-drying process between rain showers. This is a backward recurrence time process that corresponds to the renewal process of instances of rain. If a bacterium moves according to another process h, then the fact that h(t) stays above X t − means that the bacterium never hits a dry patch of soil and so survives. We describe a critical behaviour of h that separates the cases when survival is possible with a positive probability from the cases when this probability vanishes. An explicit formula for the survival probability is obtained in case h is linear and rain showers follow a Poisson process.

Stephens and Donnelly (2000) constructed an efficient sequential importance-sampling proposal distribution on coalescent histories of a sample of genes for computing the likelihood of a type configuration of genes in the sample. In the current paper a characterization of their importance-sampling proposal distribution is given in terms of the diffusion-process generator describing the distribution of the population gene frequencies. This characterization leads to a new technique for constructing importance-sampling algorithms in a much more general framework when the distribution of population gene frequencies follows a diffusion process, by approximating the generator of the process.

We study geometric moment contracting properties of nonlinear time series that are expressed in terms of iterated random functions. Under a Dini-continuity condition, a central limit theorem for additive functionals of such systems is established. The empirical processes of sample paths are shown to converge to Gaussian processes in the Skorokhod space. An exponential inequality is established. We present a bound for joint cumulants, which ensures the applicability of several asymptotic results in spectral analysis of time series. Our results provide a vehicle for statistical inferences for fractals and many nonlinear time series models.

We investigate the Russian option with a finite time horizon in the standard Black–Scholes model. The value of the option is shown to be a solution of a certain parabolic free boundary problem, and the optimal stopping boundary is shown to be continuous. Moreover, the asymptotic behavior of the optimal stopping boundary near expiration is studied.

In this paper, we consider a failure point process related to the Markovian arrival process defined by Neuts. We show that it converges in distribution to a homogeneous Poisson process. This convergence takes place in the context of rare occurrences of failures. We also provide a convergence rate of the convergence in total variation of this point process using an approach developed by Kabanov, Liptser and Shiryaev for the doubly stochastic Poisson process driven by a finite Markov process.

We drive a car along a street towards our destination and look for an available parking place without turning around. Each parking place is associated with a loss which decreases with the distance of the parking place from our destination. Assume that the states (empty or filled) of the parking places form a Markov chain. We want to find an optimal parking strategy to minimize the expected loss. A curious example is constructed and two sufficient conditions for the existence of the threshold-type optimal parking strategy are given.

The full-information best-choice problem, as posed by Gilbert and Mosteller in 1966, asks us to find a stopping rule which maximizes the probability of selecting the largest of a sequence of n i.i.d. standard uniform random variables. Porosiński, in 1987, replaced a fixed n by a random N, uniform on {1,2,…,n} and independent of the observations. A partial-information problem, imbedded in a 1980 paper of Petruccelli, keeps n fixed but allows us to observe only the sequence of ranges (max - min), as well as whether or not the current observation is largest so far. Recently, Porosiński compared the solutions to his and Petruccelli's problems and found that the two problems have identical optimal rules as well as risks that are asymptotically equal. His discovery prompts the question: why? This paper gives a good explanation of the equivalence of the optimal rules. But even under the lens of a planar Poisson process model, it leaves the equivalence of the asymptotic risks as somewhat of a mystery. Meanwhile, two other problems have been shown to have the same limiting risks: the full-information problem with the (suboptimal) Porosiński-Petruccelli stopping rule, and the full-information ‘duration of holding the best’ problem of Ferguson, Hardwick and Tamaki, which turns out to be nothing but the Porosiński problem in disguise.

A gambler, with an initial fortune less than 1, wants to buy a house which sells today for 1. Due to inflation, the price of the house tomorrow will be 1 + α, where α is a nonnegative constant, and will continue to go up at this rate, becoming (1 + α) n on the nth day. Once each day, he can stake any amount of fortune in his possession, but no more than he possesses, on a primitive casino. It is well known that, in a subfair primitive casino without the presence of inflation, the gambler should play boldly. The presence of inflation would motivate the gambler to recognize the time value of his fortune and to try to reach his goal as quickly as possible; intuitively, we would conjecture that the gambler should again play boldly. However, in this note we will show that, unexpectedly, bold play is not necessarily optimal.

De Iorio and Griffiths (2004) developed a new method of constructing sequential importance-sampling proposal distributions on coalescent histories of a sample of genes for computing the likelihood of a type configuration of genes in the sample by simulation. The method is based on approximating the diffusion-process generator describing the distribution of population gene frequencies, leading to an approximate sample distribution and finally to importance-sampling proposal distributions. This paper applies that method to construct an importance-sampling algorithm for computing the likelihood of samples of genes in subdivided population models. The importance-sampling technique of Stephens and Donnelly (2000) is thus extended to models with a Markov chain mutation mechanism between gene types and migration of genes between subpopulations. An algorithm for computing the likelihood of a sample configuration of genes from a subdivided population in an infinitely-many-alleles model of mutation is derived, extending Ewens's (1972) sampling formula in a single population. Likelihood calculation and ancestral inference in gene trees constructed from DNA sequences under the infinitely-many-sites model are also studied. The Griffiths-Tavaré method of likelihood calculation in gene trees of Bahlo and Griffiths (2000) is improved for subdivided populations.

In this paper we derive some of the main ergodicity properties of a class of Markov renewal processes and the associated marked point processes. This class represents a generic model of applied probability and is of importance in earthquake modeling, reliability theory and queueing.

The focus of our attention is the limit distribution of the sum of independent and identically distributed random vectors from which all the extreme summands are removed. The problem is rather trivial if the summands are ordered by their norms. It is of much more interest when the vertices of the convex hull generated by the vectors are taken as the extremes.

Optimal stopping rules are developed for the correlated random walk when future returns are discounted by a constant factor per unit time. The optimal rule is shown to be of dual threshold form: one threshold for stopping after an up-step, and another for stopping after a down-step. Precise expressions for the thresholds are given for both the positively and the negatively correlated cases. The optimal rule is illustrated by several numerical examples.

We consider some generalizations of the germ-grain growing model studied by Daley, Mallows and Shepp (2000). In this model, a realization of a Poisson process on a line with points Xi is fixed. At time zero, simultaneously at each Xi, a circle (grain) starts growing at the same speed. It grows until it touches another grain, and then it stops. The question is whether the point zero is eventually covered by some circle. In our note we expand this model in the following three directions. We study: a one-sided growth model with a fixed number of circles; a grain-growth model on a regular tree; and a grain-growth model on a line with non-Poisson distributed centres of the circles.

This paper considers a class of continuous-time long-range-dependent Gaussian processes. The corresponding spectral density is assumed to have a general and flexible form, which covers some important and special cases. For example, the spectral density of a continuous-time fractional stochastic differential equation is included. A modelling procedure is then established through estimating the parameters involved in the spectral density by using an extended continuous-time version of the Gauss–Whittle objective function. The resulting estimates are shown to be strongly consistent and asymptotically normal. An application of the modelling procedure to the identification and modelling of a fractional stochastic volatility is discussed in some detail.

Let Β1, Β2 be a pair of Banach spaces and T be a vector valued martingale transform (with respect to general filtration) which maps Β1-valued martingales into Β2-valued martingales. Then, the following statements are equivalent: T is bounded from into for some p (or equivalently for every p) in the range 1 < p < ∞; T is bounded from into BMOB2; T is bounded from BMOB1 into BMOB2; T is bounded from into . Applications to UMD and martingale cotype properties are given. We also prove that the Hardy space defined in the case of a general filtration has nice dense sets and nice atomic decompositions if and only if Β has the Radon-Nikodým property.

We consider a processing system, composed of several parallel queues and a processor, which operates in a time-varying environment that fluctuates between various states or modes. The service rate at each queue depends on the processor bandwidth allocated to it, as well as the environment mode. Each queue is driven by a job traffic flow, which may also depend on the environment mode. Dynamic processor scheduling policies are investigated for maximizing the system throughput, by adapting to queue backlogs and the environment mode. We show that allocating the processor bandwidth to the queues, so as to maximize the projection of the service rate vector onto a linear function of the workload vector, can keep the system stable under the maximum possible traffic load. The analysis of the system dynamics is first done under very general assumptions, addressing rate stability and flow conservation on individual traffic and environment evolution traces. The connection with stochastic stability is later discussed for stationary and ergodic traffic and environment processes. Various extensions to feed-forward networks of such nodes, the multi-processor case, etc., are also discussed. The approach advances the methodology of trace-based modelling of queueing structures. Applications of the model include bandwidth allocation in wireless channels with fluctuating interference and allocation of switching bandwidth to traffic flows in communication networks with fluctuating congestion levels.

The maximum drawdown at time T of a random process on [0,T] can be defined informally as the largest drop from a peak to a trough. In this paper, we investigate the behaviour of this statistic for a Brownian motion with drift. In particular, we give an infinite series representation of its distribution and consider its expected value. When the drift is zero, we give an analytic expression for the expected value, and for nonzero drift, we give an infinite series representation. For all cases, we compute the limiting (T → ∞) behaviour, which can be logarithmic (for positive drift), square root (for zero drift) or linear (for negative drift).

We observe a sequence X 1, X 2,…, X n of independent and identically distributed coordinatewise nonnegative d-dimensional random vectors. When a vector is observed it can either be selected or rejected but once made this decision is final. In each coordinate the sum of the selected vectors must not exceed a given constant. The problem is to find a selection policy that maximizes the expected number of selected vectors. For a general absolutely continuous distribution of the X i we determine the maximal expected number of selected vectors asymptotically and give a selection policy which asymptotically achieves optimality. This problem raises a question closely related to the following problem. Given an absolutely continuous measure μ on Q = [0,1] d and a τ ∈ Q, find a set A of maximal measure μ(A) among all A ⊂ Q whose center of gravity lies below τ in all coordinates. We will show that a simplicial section { x ∈ Q | 〈 x , θ 〉 ≤ 1}, where θ ∈ ℝ d , θ ≥ 0, satisfies a certain additional property, is a solution to this problem.

The aim of this paper is to analyze the probabilistic structure for a rather general class of bilinear models systematically. First, the sufficient and necessary conditions for stationarity are given with a concise expression. Then both the autocovariance function and the spectral density function are obtained. The Yule–Walker-type difference equations for autocovariances are derived by means of the spectral density function. Concerning the second-order probabilistic structure, the model is similar to an ARMA model. The third-order probabilistic structure for the model is discussed and a group of Yule–Walker-type difference equations for third-order cumulants are discovered.

We study estimation of the tail-decay parameter of the marginal distribution corresponding to a discrete-time, real-valued stationary stochastic process. Assuming that the underlying process is short-range dependent, we investigate properties of estimators of the tail-decay parameter which are based on the maximal extreme value of the process observed over a sampled time interval. These estimators only assume that the tail of the marginal distribution is roughly exponential, plus some modest ‘mixing’ conditions. Consistency properties of these estimators are established, as well as minimax convergence rates. We also provide some discussion on estimating the pre-exponent, when a more refined tail asymptotic is assumed. Properties of a certain moving-average variant of the extremal-based estimator are investigated as well. In passing, we also characterize the precise dependence (mixing) assumptions that support almost-sure limit theory for normalized extreme values and related first-passage times in stationary sequences.