In this paper we study the asymptotic relationship between the loss ratio in a finite buffer system and the overflow probability (the tail of the queue length distribution) in the corresponding infinite buffer system. We model the system by a fluid queue which consists of a server with constant rate c and a fluid input. We provide asymptotic upper and lower bounds on the difference between log P{Q > x} and logPL(x) under different conditions. The conditions for the upper bound are simple and are satisfied by a very large class of input processes. The conditions on the lower bound are more complex but we show that various classes of processes such as Markov modulated and ARMA type Gaussian input processes satisfy them.

We consider a parametrization of the Heath-Jarrow-Morton (HJM) family of term structure of interest rate models that allows a finite-dimensional Markovian representation of the stochastic dynamics. This parametrization results from letting the volatility function depend on time to maturity and on two factors: the instantaneous spot rate and one fixed-maturity forward rate. Our main purpose is an estimation methodology for which we have to model the observations under the historical probability measure. This leads us to consider as an additional third factor the market price of interest rate risk, that connects the historical and the HJM martingale measures. Assuming that the information comes from noisy observations of the fixed-maturity forward rate, the purpose is to estimate recursively, on the basis of this information, the three Markovian factors as well as the parameters in the model, in particular those in the volatility function. This leads to a nonlinear filtering problem, for the solution of which we describe an approximation methodology, based on time discretization and quantization. We prove the convergence of the approximate filters for each of the observed trajectories.

Seeds are randomly scattered in ℝd according to an m-dependent point process. Each seed has its own potential germination time. From each seed that succeeds in germinating, a spherical inhibited region grows to prohibit germination of any seed with later potential germination time. We show that under certain conditions on the distribution of the potential germination time, the number of germinated seeds in a large region has an asymptotic normal distribution.

Let n points be placed uniformly at random in a subset A of the plane. A point is said to be maximal in the configuration if no other point is larger in both coordinates. We show that, for large n and for many sets A, the number of maximal points is approximately normally distributed. The argument uses Stein's method, and is also applicable in higher dimensions.

This article deals with the distribution of the view of a random environment as seen by an observer whose location at each moment is determined by the environment. The main application is in statistical fluid mechanics, where the environment consists of a random velocity field and the observer is a particle moving in the velocity field, possibly subject to molecular diffusion. Several results on such Lagrangian observations of the environment have appeared in the literature, beginning with the 1957 dissertation of J. L. Lumley. This article unites these results into a simple unified framework and rounds out the theory with new results in several directions. When the environment is homogeneous, the problem can be re-cast in terms of certain random mappings on the physical space that are based on the random location of the observer. If these mappings preserve the invariant measure on the physical space, then the view from the random location has the same distribution as the view from the origin. If these mappings satisfy the flow property and the environment is stationary, then the succession of Lagrangian observations over time forms a strictly stationary process. In particular, for motion in a homogeneous, stationary, and nondivergent velocity field, the Lagrangian velocity (the velocity of the particle) is strictly stationary, which was first observed by Lumley. In the compressible case, the distribution of a Lagrangian observation has a density with respect to the distribution of the view from the origin, and in some cases convergence in distribution of the Lagrangian observations as time tends to infinity can be shown.

A complete and sufficient statistic is found for various stationary Poisson processes of compact sets with known primary grain. In the particular case of a segment process, the uniformly best unbiased estimator for the length density is the number of segments hitting the sampling window divided by a certain constant and multiplied by the mean segment length.

Connections between classification and lumpability in the stochastic Hopfield model (SHM) are explored and developed. A simplification of the SHM's complexity based upon its inherent lumpability is derived. Contributions resulting from this reduction in complexity include: (i) computationally feasible classification time computations; (ii) a development of techniques for enumerating the stationary distribution of the SHM's energy function; and (iii) a characterization of the set of possible absorbing states of the Markov chain associated with the zero temperature SHM.

We define the extension of the so-called ‘martingales in the branching random walk’ in R or C to some Banach algebras B of infinite dimension and give conditions for their convergence, almost surely and in the Lp norm. This abstract approach gives conditions for the simultaneous convergence of uncountable families of such martingales constructed simultaneously in C, the idea being to consider such a family as a function-valued martingale in a Banach algebra of functions. The approach is an alternative to those of Biggins (1989), (1992) and Barral (2000), and it applies to a class of families to which the previous approach did not. We also give a result on the continuity of these multiplicative processes. Our results extend to a varying environment version of the usual construction: instead of attaching i.i.d. copies of a given random vector to the nodes of the tree ∪n≥0N+n, the distribution of the vector depends on the node in the multiplicative cascade. In this context, when B=R and in the nonnegative case, we generalize the measure on the boundary of the tree usually related to the construction; then we evaluate the dimension of this nonstatistically self-similar measure. In the self-similar case, our convergence results make it possible to simultaneously define uncountable families of such measures, and then to estimate their dimension simultaneously.

This paper raises the following question: let {Φn(A), A ⊂ ℝd} be a Poisson process with intensity nf(x), x ∈ ℝd and let c(Xi | Φn) be a Voronoi tile with nucleus Xi (a jump point of Φn). Let μ(.) denote Lebesgue measure in ℝd. Is it true that, for any bounded measurable subset B of ℝd, ∑Xi∈Bμ(c(Xi| Φn)) → μ(B) almost surely as n → ∞ only if f > 0 almost everywhere? This statement can be viewed as the strong law of large numbers for Voronoi tessellation. Though the positive answer may seem ‘obvious’, we could not find any such statement, especially for arbitrary measurable B and nonhomogeneous Poisson processes. For B with the boundary of Lebesgue measure 0 the proof is simple. We prove in this paper that the statement is true for ℝ1.

We prove that for a class of anisotropic long-range percolation models for which connection probabilities p<x,z> satisfy some regularity properties, and such that ∑z∈Z2p<x,z> = ∞, percolation still will occur even if we truncate all edges whose length exceeds some constant (which in this case depends on the family of connectivity probabilities {p<x,z>). We also present an example of dependent long-range percolation model for which this is not true.

We analyse a non-Markovian generalization of the telegrapher's random process. It consists of a stochastic process describing a motion on the real line characterized by two alternating velocities with opposite directions, where the random times separating consecutive reversals of direction perform an alternating renewal process. In the case of Erlang-distributed interrenewal times, explicit expressions of the transition densities are obtained in terms of a suitable two-index pseudo-Bessel function. Some results on the distribution of the maximum of the process are also disclosed.

In this paper, we present the distribution of the coalescence time of two DNA sequences (or genes) subject to symmetric migration between two islands, and conditional on the observed number of segregating sites in the sequences. The distribution for the segregating-site pattern is also obtained. Some surprising results emerge when both genes are initially on the same island. First, the post-data mean coalescence time is shown to be dependent on the migration parameter, as opposed to the pre-data mean. Second, both the post-data density and expectation for the coalescence time are shown to converge, in the weak-migration limit, to the corresponding panmictic results, as opposed to the pre-data situation where there is convergence in the density but not in the expectation. Finally, it is shown that there is convergence in the weak-migration limit in the distribution of the number of segregating sites but not in the expectation and variance. Numerical and graphical results for samples of size greater than two are also presented.

The distribution of the length of a typical chord of a stationary random set is an interesting feature of the set's whole distribution. We give a nonparametric estimator of the chord length distribution and prove its strong consistency. We report on a simulation experiment in which our estimator compared favourably to a reduced sample estimator. Both estimators are illustrated by applying them to an image sample from a yoghurt ferment. We briefly discuss the closely related problem of estimation of the linear contact distribution. We show by a simulation experiment that a transformation of our estimator of the chord length distribution is more efficient than a Kaplan-Meier type estimator of the linear contact distribution.

The paper describes a new numerical method for the calculation of noncrossing probabilities for arbitrary boundaries by a Poisson process. We find the method to be simple in implementation, quick and efficient - it works reliably for Poisson processes of very high intensity n, up to several thousand. Hence, it can be used to detect unusual features in the finite-sample behaviour of empirical process and trace it down to very high sample sizes. It also can be used as a good approximation for noncrossing probabilities for Brownian motion and Brownian bridge, in particular when the boundaries are not regular. As a numerical example we demonstrate the divergence of normalized Kolmogorov-Smirnov statistics from their prescribed limiting distributions (Eicker (1979), Jaeshke (1979)) for quite large n in contrast to very regular behaviour of statistics of Mason (1983). For the Brownian motion case we considered square-root, Daniels' (1969) and Grooneboom's (1989) boundaries.

A network is a system of segments or edges in ℝd which intersect only in the segment endpoints, which are called vertices. An example is the system of edges of a tessellation. It is possible to give formulas for the specific connectivity number of a random network; in the stationary case, the intensity of the 0-curvature measure is equal to the difference of the intensities of the point processes of vertices and edge centres.

Sharp upper and lower bounds are derived for the solution of renewal equations. These include as special cases exponential inequalities, some of which have been derived for specific renewal equations. Together with the well-known Cramér-Lundberg asymptotic estimate, these bounds give additional information about the behaviour of the solution. Nonexponential bounds, which are of use in connection with defective renewal equations, are also obtained. The results are then applied in examples involving the severity of insurance ruin, age-dependent branching processes, and a generalized type II Geiger counter.