Showing papers in "Advances in Applied Probability in 1999"

Generalized gamma measures and shot-noise Cox processes

Abstract: 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.

Asymptotic results for multiplexing subexponential on-off processes

Abstract: Consider an aggregate arrival process A N obtained by multiplexing N on-off processes with exponential off periods of rate λ and subexponential on periods τ on .A sN goes to infinity, with λN → � , A N approaches an M/G/∞ type process. Both for finite and infinite N , we obtain the asymptotic characterization of the arrival process activity period. Using these results we investigate a fluid queue with the limiting M/G/∞ arrival process A ∞ and capacity c. When on periods are regularly varying (with non-integer exponent), we derive a precise asymptotic behavior of the queue length random variable Q P observed at the beginning of the arrival process activity periods ∞ x/(r+ρ−c) [τ on > u] d ux →∞ , where ρ = A ∞ < c; r (c ≤ r) is the rate at which the fluid is arriving during an on period. The asymptotic (time average) queue distribution lower bound is obtained under more general assumptions regarding on periods than regular variation. In addition, we analyse a queueing system in which one on-off process, whose on period belongs to a subclass of subexponential distributions, is multiplexed with independent exponential processes with aggregate expected rate et . This system is shown to be asymptotically equivalent to the same queueing system with the exponential arrival processes being replaced by their total mean value et .

The size of the connected components of excursion sets of χ2, t and F fields

Abstract: 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.

Recent common ancestors of all present-day individuals

Abstract: Previous study of the time to a common ancestor of all present-day individuals has focused on models in which each individual has just one parent in the previous generation. For example, ‘mitochondrial Eve’ is the most recent common ancestor (MRCA) when ancestry is defined only through maternal lines. In the standard Wright-Fisher model with population size n, the expected number of generations to the MRCA is about 2n, and the standard deviation of this time is also of order n. Here we study a two-parent analog of the Wright-Fisher model that defines ancestry using both parents. In this model, if the population size n is large, the number of generations, 𝒯 n , back to a MRCA has a distribution that is concentrated around lgn (where lg denotes base-2 logarithm), in the sense that the ratio 𝒯 n (lgn) converges in probability to 1 as n→∞. Also, continuing to trace back further into the past, at about 1.77 lgn generations before the present, all partial ancestry of the current population ends, in the following sense: with high probability for large n, in each generation at least 1.77lgn generations before the present, all individuals who have any descendants among the present-day individuals are actually ancestors of all present-day individuals.

Reaching Goals by a deadline: digital options and continuous-time active portfolio management

Abstract: 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 ,f or 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.

Tail probabilities for non-standard risk and queueing processes with subexponential jumps

Abstract: A well-known result on the distribution tail of the maximum of a random walk with heavy-tailed increments is extended to more general stochastic processes. Results are given in different settings, involving, for example, stationary increments and regeneration. Several examples and counterexamples illustrate that the conditions of the theorems can easily be verified in practice and are in part necessary. The examples include superimposed renewal processes, Markovian arrival processes, semi-Markov input and Cox processes with piecewise constant intensities.

Quermass-interaction processes: conditions for stability

Abstract: We consider a class of random point and germ-grain processes, obtained using a rather natural weighting procedure. Given a Poisson point process, on each point one places a grain, a (possibly random) compact convex set. Let Ξ be the union of all grains. One can now construct new processes whose density is derived from an exponential of a linear combination of quermass functionals of Ξ. If only the area functional is used, then the area-interaction point process is recovered. New point processes arise if we include the perimeter length functional, or the Euler functional (number of components minus number of holes). The main question addressed by the paper is that of when the resulting point process is well-defined: geometric arguments are used to establish conditions for the point process to be stable in the sense of Ruelle.

Optimal stochastic scheduling of a two-stage tandem queue with parallel servers

Abstract: We consider the optimal stochastic scheduling of a two-stage tandem queue with two parallel servers. The servers can serve either queue at any point in time and the objective is to minimize the total holding costs incurred until all jobs leave the system. We characterize sufficient and necessary conditions under which it is optimal to allocate both servers to the upstream or downstream queue. We then conduct a numerical study to investigate whether the results shown for the static case also hold for the dynamic case. Finally, we provide a numerical study that explores the benefits of having two flexible parallel servers which can work at either queue versus servers dedicated to each queue. We discuss the results' implications for cross-training workers to perform multiple tasks.

Perfect simulation of some point processes for the impatient user

Abstract: Recently Propp and Wilson [14] have proposed an algorithm, called coupling from the past (CFTP), which allows not only an approximate but perfect (i.e. exact) simulation of the stationary distribution of certain finite state space Markov chains. Perfect sampling using CFTP has been successfully extended to the context of point processes by, amongst other authors, Haggstrom et al. [5]. In [5] Gibbs sampling is applied to a bivariate point process, the penetrable spheres mixture model [19]. However, in general the running time of CFTP in terms of number of transitions is not independent of the state sampled. Thus an impatient user who aborts long runs may introduce a subtle bias, the user impatience bias. Fill [3] introduced an exact sampling algorithm for finite state space Markov chains which, in contrast to CFTP, is unbiased for user impatience. Fill's algorithm is a form of rejection sampling and similarly to CFTP requires sufficient monotonicity properties of the transition kernel used. We show how Fill's version of rejection sampling can be extended to an infinite state space context to produce an exact sample of the penetrable spheres mixture process and related models. Following [5] we use Gibbs sampling and make use of the partial order of the mixture model state space. Thus we construct an algorithm which protects against bias caused by user impatience and which delivers samples not only of the mixture model but also of the attractive area-interaction and the continuum random-cluster process.

Asymptotics of Poisson approximation to random discrete distributions: an analytic approach

Abstract: A general analytic scheme for Poisson approximation to discrete distributions is studied in which the asymptotic behaviours of the generalized total variation, Fortet-Mourier (or Wasserstein), Kolmogorov and Matusita (or Hellinger) distances are explicitly characterized. Applications of this result include many number-theoretic functions and combinatorial structures. Our approach differs from most of the existing ones in the literature and is easily amended for other discrete approximations; arithmetic and combinatorial examples for Bessel approximation are also presented. A unified approach is developed for deriving uniform estimates for probability generating functions of the number of components in general decomposable combinatorial structures, with or without analytic continuation outside their circles of convergence.

Interacting particle systems approximations of the Kushner-Stratonovitch equation

Abstract: In this paper we consider the continuous-time filtering problem and we estimate the order of convergence of an interacting particle system scheme presented by the authors in previous works We will discuss how the discrete time approximating model of the Kushner-Stratonovitch equation and the genetic type interacting particle system approximation combine We present quenched error bounds as well as mean order convergence results

Stopping sets: Gamma-type results and hitting properties

Abstract: Recently in the paper \cite{MolZuy:96} there was established the following Gamma-type result. Given the number $N$ of a homogeneous Poisson process' points defining a random figure, its volume is $\Gamma(N,\lambda)$ distributed, where $\lambda$ is the intensity of the process. The goal of this paper is to give an alternative description of the class of the random sets for which the Gamma-type results hold. We show that it corresponds to the class of \emph{stopping sets} with respect to the natural filtration of the point process with certain scaling properties. The proof is very short and uses the martingale technique for directed processes, in particular, the analog of the Doob's optional sampling theorem proved in \cite{Kur:80}. Along with an elegance, this approach provides a new inside into the nature of geometrical objects constructed with respect to a point process. We show, in particular, that in the Poisson case the probability of a point to be covered by a stopping set does not depend on whether it is point of the Poisson process or not.

Asymptotics of palm-stationary buffer content distributions in fluid flow queues

Abstract: We study a fluid flow queueing system with m independent sources alternating between periods of silence and activity; m ≥ 2. The distribution function of the activity periods of one source, is supposed to be intermediate regular varying . We show that the distribution of the net increment of the buffer during an aggregate activity period (i.e. when at least one source is active) is asymptotically tail-equivalent to the distribution of the net input during a single activity period with intermediate regular varying distribution function. In this way, we arrive at an asymptotic representation of the Palm-stationary tail-function of the buffer content at the beginning of aggregate activity periods. Our approach is probabilistic and extends recent results of Boxma (1996; 1997) who considered the special case of regular variation.

Existence of 'nearest-neighbour' spatial gibbs models

Abstract: The present study deals with the existence of ‘nearest-neighbour’ type Gibbs models, introduced by Baddeley and Moller in 1989. In such models, the neighbourhood relation depends on the realization of the process. After giving new sufficient conditions to prove the existence of stationary Gibbs states, we deal with the first-nearest-neighbour model, the triplets Delaunay model, Ord's model and Markov connected component type models.

Reduced subcritical galton-watson processes in a random environment

Abstract: We study the structure of genealogical trees of reduced subcritical Galton-Watson processes in a random environment assuming that all (randomly varying in time) offspring generating functions are fractional linear. We show that this structure may differ significantly from that of the ‘classical’ reduced subcritical Galton-Watson processes. In particular, it may look like a complex ‘hybrid’ of classical reduced super and subcritical processes. Some relations with random walks in a random environment are discussed.

The maximum of a function of a Markov chain and application to linkage analysis

Abstract: One method of linkage analysis in humans is based on identity-by-descent of pairs of relatives who share a phenotype of interest (for example, a particular disease). We replace the convenient assumption of continuous specification of regions of identity by descent by the more realistic, although still artificially simple, assumption of data from a discrete set of equally spaced infinitely polymoxphic markers. We generalize the continuous time Markov chain analysis of Feingold (1993b) and compare the accuracy of the new approximation with that of the simpler Gaussian approximation of Feingold, Brown and Siegmund (1993) undera variety of assumptions about the composition ofthe pedigrees to be studied. We also suggest a perturbation of the Gaussian approximation as a compromise to achieve reasonable accuracy with minimal computational effort.

The fundamental matrix of singularly perturbed Markov chains

Abstract: We consider a singularly perturbed (finite state) Markov chain and provide a complete characterization of the fundamental matrix. In particular, we obtain a formula for the regular part simpler than a previous formula obtained by Schweitzer, and the singular part is obtained via a reduction process similar to Delebecque's reduction for the stationary distribution. In contrast to previous approaches, one works with aggregate Markov chains of much smaller dimension than the original chain, an essential feature for practical computation. An application to mean first-passage times is also presented.

On the size of a random sphere of influence graph

Abstract: We approach sphere of influence graphs (SIGs) from a probabilistic perspective. Ordinary SIGs were first introduced by Toussaint as a type of proximity graph for use in pattern recognition, computer vision and other low-level vision tasks. A random sphere of influence graph (RSIG) is constructed as follows. Consider n points uniformly and independently distributed within the unit square in d dimensions. Around each point, X i , draw an open ball (‘sphere of influence’) with radius equal to the distance to X i 's nearest neighbour. Finally, draw an edge between two points if their spheres of influence intersect. Asymptotically exact values for the expected number of edges in a RSIG are determined for all values of d ; previously, just upper and lower bounds were known for this quantity. A modification of the Azuma-Hoeffding exponential inequality is employed to exhibit the sharp concentration of the number of edges around its expected value.

The central limit theorem for euclidean minimal spanning trees ii

Abstract: Let X i : i ≥ 1 be i.i.d. points in ℝ d , d ≥ 2, and let T n be a minimal spanning tree on X 1 ,…, X n . Let L ( X 1 ,…, X n ) be the length of T n and for each strictly positive integer α let N ( X 1 ,…, X n ;α) be the number of vertices of degree α in T n . If the common distribution satisfies certain regularity conditions, then we prove central limit theorems for L ( X 1 ,…, X n ) and N ( X 1 ,…, X n ;α). We also study the rate of convergence for E L ( X 1 ,…, X n ).

First contact distributions for spatial patterns: regularity and estimation

Abstract: For applications in spatial statistics an important property of a random set X in Rk is its rst contact distribution This is the distribution of the distance from a xed point to the nearest point of X where distance is measured using scalar dilations of a xed test set B We show that if B is convex and contains a neighbourhood of the rst contact distribution function FB is absolutely continuous We give two explicit representations of FB and additional regularity conditions under which FB is continuously dierentiable A KaplanMeier estimator of FB is introduced and its basic properties examined

Stochastic comparisons for residual lifetimes and Bayesian notions of multivariate ageing

Abstract: We compare distributions of residual lifetimes of dependent components of different age. This approach yields several notions of multivariate ageing. A special feature of our notions is that they are based on one-dimensional stochastic comparisons. Another difference from the traditional approach is that we do not condition on different histories.

On the supremum distribution of integrated stationary Gaussian processes with negative linear drift

Abstract: In this paper we study the supremum distribution of a class of Gaussian processes having stationary increments and negative drift using key results from Extreme Value Theory. We focus on deriving an asymptotic upper bound to the tail of the supremum distribution of such processes. Our bound is valid for both discrete- and continuous-time processes. We discuss the importance of the bound, its applicability to queueing problems, and show numerical examples to illustrate its performance.

The volume fraction of a poisson germ model with maximally non-overlapping spherical grains

Abstract: This paper considers a germ-grain model for a random system of non-overlapping spheres in Rd for d = 1, 2 and 3. The centres of the spheres (i.e. the 'germs' for the 'grains') form a stationary Poisson process; the spheres result from a uniform growth process which starts at the same instant in all points in the radial direction and stops for any sphere when it touches any other sphere. Upper and lower bounds are derived for the volume fraction of space occupied by the spheres; simulation yields the values 0.632, 0.349 and 0.186 for d = 1, 2 and 3. The simulations also provide an estimate of the tail of the distribution function of the volume of a randomly chosen sphere; these tails are compared with those of two exponential distributions, of which one is a lower bound and is an asymptote at the origin, and the other has the same mean as the simulated distribution. An upper bound on the tail of the distribution is also an asymptote at the origin but has a heavier tail than either of these exponential distributions. More detailed information for the one-dimensional case has been found by Daley, Mallows and Shepp; relevant information is summarized, including the volume fraction 1 - e-1 = 0.632 12 and the tail of the grain volume distribution e-Y exp(e-Y - 1), which is closer to the simulated tails for d = 2 and 3 than the exponential bounds.

RANDOM p-CONTENT OF A p-PARALLELOTOPE IN EUCLIDEAN n-SPACE

Abstract: Techniques currently available in the literature in dealing with problems in geometric probabilities seem to rely heavily on results from differential and integral geometry. This paper provides a radical departure in this respect. By using purely algebraic procedures and making use of some properties of Jacobians of matrix transformations and functions of matrix argument, the distributional aspects of the random p-content of a p-parallelotope in Euclidean n-space are studied. The common assumptions of independence and rotational invariance of the random points are relaxed and the exact distributions and arbitrary moments, not just integer moments, are derived in this article. General real matrix-variate families of distributions, whose special cases include the mulivariate Gaussian, a multivariate type-l beta, a multivariate type-2 beta and spherically symmetric distributions, are considered.

Incomplete markets: convergence of options values under the minimal martingale measure

Abstract: In the setting of incomplete markets, this paper presents a general result of convergence for derivative assets prices. It is proved that the minimal martingale measure first introduced by Follmer and Schweizer is a convenient tool for the stability under convergence. This extends previous well-known results when the markets are complete both in discrete time and continuous time. Taking into account the structure of stock prices, a mild assumption is made. It implies the joint convergence of the sequences of stock prices and of the Radon-Nikodym derivative of the minimal measure. The convergence of the derivatives prices follows. This property is illustrated in the main classes of financial market models.

Multi-resolution approximation to the stochastic inverse problem

Abstract: The linear inverse problem of estimating the input random field in a first-kind stochastic integral equation relating two random fields is considered. For a wide class of integral operators, which includes the positive rational functions of a self-adjoint elliptic differential operator on L 2(ℝ d ), the ill-posed nature of the problem disappears when such operators are defined between appropriate fractional Sobolev spaces. In this paper, we exploit this fact to reconstruct the input random field from the orthogonal expansion (i.e. with uncorrelated coefficients) derived for the output random field in terms of wavelet bases, transformed by a linear operator factorizing the output covariance operator. More specifically, conditions under which the direct orthogonal expansion of the output random field coincides with the integral transformation of the orthogonal expansion derived for the input random field, in terms of an orthonormal wavelet basis, are studied.

ISOTROPIC CORRELATION FUNCTIONS ON d-DIMENSIONAL BALLS

Abstract: A popular procedure in spatial data analysis is to fit a line segment of the form c(x) = 1 - a llx II, Ixll < 1, to observed correlations at (appropriately scaled) spatial lag x in d-dimensional space. We show that such an approach is permissible if and only if

Large deviations, moderate deviations, and queues with long-range dependent input

Abstract: Long-range dependence has been recently asserted to be an important characteristic in modeling telecommunications traffic. Inspired by the integral relationship between the fractional Brownian motion and the standard Brownian motion, we model a process with long-range dependence, Y, as a fractional integral of Riemann-Liouville type applied to a more standard process X-one that does not have long-range dependence. VVhen X takes the form of a sample path process with bounded stationary increments, we provide a criterion for X to satisfy a moderate deviations principle (MDP). Based on the MDP of X, we then establish the MDP for Y. Furthermore, we characterize, in terms of the MDP, the transient behavior of queues when fed with the long-range dependent input process Y. In particular, we identify the most likely path that leads to a large queue, and demonstate that unlike the case where the input has short-range dependence, the path here is nonlinear.

Weak approximations for empirical lorenz curves and their goldie inverses of stationary observations

Abstract: By using Chibisov-O'Reilly type theorems for uniform empirical and quantile processes based on stationary observations, we establish a weak approximation theory for empirical Lorenz curves and their inverses used in economics. In particular, we obtain weak approximations for empirical Lorenz curves and their inverses also under the assumptions of mixing dependence, often used structures of dependence for observations.

Stochastic analysis of image acquisition, interpolation and scale-space smoothing

Abstract: In the high-level operations of computer vision it is taken for granted that image features have been reliably detected. This paper addresses the problem of feature extraction by scale-space methods. There has been a strong development in scale-space theory and its applications to low-level vision in the last couple of years. Scale-space theory for continuous signals is on a firm theoretical basis. However, discrete scale-space theory is known to be quite tricky, particularly for low levels of scale-space smoothing. The paper is based on two key ideas: to investigate the stochastic properties of scale-space representations and to investigate the interplay between discrete and continuous images. These investigations are then used to predict the stochastic properties of sub-pixel feature detectors. The modeling of image acquisition, image interpolation and scale-space smoothing is discussed, with particular emphasis on the influence of random errors and the interplay between the discrete and continuous representations. In doing so, new results are given on the stochastic properties of discrete and continuous random fields. A new discrete scale-space theory is also developed. In practice this approach differs little from the traditional approach at coarser scales, but the new formulation is better suited for the stochastic analysis of sub-pixel feature detectors. The interpolated images can then be analysed independently of the position and spacing of the underlying discretisation grid. This leads to simpler analysis of sub-pixel feature detectors. The analysis is illustrated for edge detection and correlation. The stochastic model is validated both by simulations and by the analysis of real images.