Showing papers in "Annals of Applied Probability in 1997"

Weak convergence and optimal scaling of random walk Metropolis algorithms

Abstract: This paper considers the problem of scaling the proposal distribution of a multidimensional random walk Metropolis algorithm in order to maximize the efficiency of the algorithm. The main result is a weak convergence result as the dimension of a sequence of target densities, n, converges to $\infty$. When the proposal variance is appropriately scaled according to n, the sequence of stochastic processes formed by the first component of each Markov chain converges to the appropriate limiting Langevin diffusion process. The limiting diffusion approximation admits a straightforward efficiency maximization problem, and the resulting asymptotically optimal policy is related to the asymptotic acceptance rate of proposed moves for the algorithm. The asymptotically optimal acceptance rate is 0.234 under quite general conditions. The main result is proved in the case where the target density has a symmetric product form. Extensions of the result are discussed.

Epidemics with two levels of mixing

Abstract: We consider epidemics with removal (SIR epidemics) in populations that mix at two levels: global and local. We develop a general modelling framework for such processes, which allows us to analyze the conditions under which a large outbreak is possible, the size of such outbreaks when they can occur and the implications for vaccination strategies, in each case comparing our results with the simpler homogeneous mixing case. More precisely, we consider models in which each infectious individual i has a global probability $p_G$ for infecting each other individual in the population and a local probability $p_L$, typically much larger, of infecting each other individual among a set of neighbors $\mathscr{N}(i)$. Our main concern is the case where the population is partitioned into local groups or households, but our approach also applies to cases where neighborhoods do not form a partition, for instance, to spatial models with a mixture of local (e.g., nearest-neighbor) and global contacts. We use a variety of theoretical approaches: a random graph framework for the initial exposition of the simple case where an individual's contacts are independent; branching process approximations for the general threshold result; and an embedding representation for rigorous results on the final size of outbreaks. From the applied viewpoint the key result is that, compared with the homogeneous mixing model in which individuals make contacts simply with probability $p_G$, the local infectious contacts have an "amplification" effect. The basic reproductive ratio of the epidemic is increased from its individual-to-individual value $R_G$ in the absence of local infections to a group-to-group value $R_* = \mu R_G$, where $\mu$ is the mean size of an outbreak, started by a randomly chosen individual, in which only local infections count. Where the groups are large and the within-group epidemics are above threshold, this amplification can permit an outbreak in the whole population at very low levels of $p_G$, for instance, for $p_G = O(1/Nn)$ in a population of N divided into groups of size n. The implication of these results for control strategies is that vaccination should be directed preferentially toward reducing $\mu$; we discuss the conditions under which the equalizing strategy, aimed at leaving unvaccinated sets of neighbors of equal sizes, is optimal. We also discuss the estimation of our threshold parameter $R_*$ from data on epidemics among households.

The longest edge of the random minimal spanning tree

Abstract: For n points placed uniformly at random on the unit square, suppose $M_n$ (respectively, $M'_n$) denotes the longest edge-length of the nearest neighbor graph (respectively, the minimal spanning tree) on these points. It is known that the distribution of $n \pi M_n^2 - \log n$ converges weakly to the double exponential; we give a new proof of this. We show that $P[M'_n = M_n] \to 1$, so that the same weak convergence holds for $M'_n$ .

Stein's method and the zero bias transformation with application to simple random sampling

Abstract: Let W be a random variable with mean zero and variance � 2 . The distribution of a variate W ∗ , satisfying EWf(W) = � 2 Ef ′ (W ∗ ) for smooth functions f, exists uniquely and defines the zero bias transformation on the distribution of W. The zero bias transformation shares many interesting properties with the well known size bias transformation for non-negative variables, but is applied to variables taking on both positive and negative values. The transformation can also be defined on more general random objects. The relation between the transformation and the expression wf ′ (w)−� 2 f ′′ (w) which appears in the Stein equation characterizing the

Exponential decay and ergodicity of completely asymmetric Lévy processes in a finite interval

Abstract: Consider a completely asymmetric Levy process which has absolutely continuous transition probabilities. We determine the exponential decay parameter $\rho$ and the quasistationary distribution for the transition probabilities of the Levy process killed as it exits from a finite interval, prove that the killed process is $\rho$-positive and specify the $\rho$-invariant function and measure.

Counterexamples in importance sampling for large deviations probabilities

Abstract: A guiding principle in the efficient estimation of rare-event probabilities by Monte Carlo is that importance sampling based on the change of measure suggested by a large deviations analysis can reduce variance by many orders of magnitude. In a variety of settings, this approach has led to estimators that are optimal in an asymptotic sense. We give examples, however, in which importance sampling estimators based on a large deviations change of measure have provably poor performance. The estimators can have variance that decreases at a slower rate than a naive estimator, variance that increases with the rarity of the event, and even infinite variance. For each example, we provide an alternative estimator with provably efficient performance. A common feature of our examples is that they allow more than one way for a rare event to occur; our alternative estimators give explicit weight to lower probability paths neglected by leading-term asymptotics.

On coupling constructions and rates in the CLT for dependent summands with applications to the antivoter model and weighted $U$-statistics

Abstract: This paper deals with rates of convergence in the CLT for certain types of dependency. The main idea is to combine a modification of a theorem of Stein, requiring a coupling construction, with a dynamic set-up provided by a Markov structure that suggests natural coupling variables. More specifically, given a stationary Markov chain $X^{(t(}$, and a function $U = U(X^{(t)})$, we propose a way to study the proximity of U to a normal random variable when the state space is large. We apply the general method to the study of two problems. In the first, we consider the antivoter chain $X^{(t)} = {X_i^{(t)}} _{i \epsilon \mathscr{V}}, t = 0, 1, \dots,$ where $\mathscr{V}$ is the vertex set of an n-vertex regular graph, and $X_i^{(t)} = +1 \text{or} -1$. The chain evolves from time t to $t + 1$ by choosing a random vertex i, and a random neighbor of it j, and setting $X_i^{(t+1)} = -X_j^{(t)}$ and $X_k^{(t+1)} = X_k^{(t)}$ for all $k ot= i$. For a stationary antivoter chain, we study the normal approximation of $U_n = U_n^{(t)} = \Sigma_i X_i^{(t)}$ for large n and consider some conditions on sequences of graphs such that $U_n$ is asymptotically normal, a problem posed by Aldous and Fill. The same approach may also be applied in situations where a Markov chain does not appear in the original statement of a problem but is constructed as an auxiliary device. This is illustrated by considering weighted U-statistics. In particular we are able to unify and generalize some results on normal convergence for degenerate weighted U-statistics and provide rates.

Dynamic control of Brownian networks: state space collapse and equivalent workload formulations

Abstract: Brownian networks are a class of linear stochastic control systems that arise as heavy traffic approximations in queueing theory. Such Brownian system models have been used to approximate problems of dynamic routing, dynamic sequencing and dynamic input control for queueing networks. A number of specific examples have been analyzed in recent years, and in each case the Brownian network has been successfully reduced to an "equivalent workload formulation" of lower dimension. In this article we explain that reduction in general terms, using an orthogonal decomposition that distinguishes between reversible and irreversible controls.

On the possibility of hedging options in the presence of transaction costs

Abstract: We study the continuous-time problem of hedging a generalized call option of the European and of the American type, in the presence of transaction costs. We show that if the price process of the relevant stock fluctuates with positive probability, then the only hedge that is possible for the American option is the trivial one. If the price of the stock, in addition to fluctuating with positive probability, is also stable with positive probability, then the same is true for the European option. We also show that in some sense, stable price with positive probability is a necessary condition for having only a trivial hedge for the European option. Our basic idea is to work with an appropriate discrete-time version of the problem which is transaction costs free. The mathematical tools that we use are elementary. A related result appears in Soner, Shreve and Cvitanic.

Patterns of buffer overflow in a class of queues with long memory in the input stream

Abstract: We study the time it takes until a fluid queue with a finite, but large, holding capacity reaches the overflow point. The queue is fed by an on/off process with a heavy tailed on distribution which is known to have long memory. It turns out that the expected time until overflow, as a function of capacity L, increases only polynomially fast; so overflows happen much more often than in the "classical" light tailed case, where the expected over-flow time increases as an exponential function of L. Moreover, we show that in the heavy tailed case overflows are basically caused by single huge jobs. An implication is that the usual $GI/G/1$ queue with finite but large holding capacity and heavy tailed service times will overflow about equally often no matter how much we increase the service rate. We also study the time until overflow for queues fed by a superposition of k iid on/off processes with a heavy tailed on distribution, and we show the benefit of pooling the system resources as far as time until overflow is concerned.

The central limit theorem for Euclidean minimal spanning trees I I

Abstract: Let ${X_i: i \geq 1}$ be i.i.d. with uniform distribution $[- 1/2, 1/2]^d, d \geq 2$, and let $T_n$ be a minimal spanning tree on ${X_1, \dots, X_n}$. For each strictly positive integer $\alpha$, let $N({X_1, \dots, X_n}; \alpha)$ be the number of vertices of degree $\alpha$ in $T_n$. Then, for each $\alpha$ such that $P(N({X_1, \dots, X_{\alpha+1}}; \alpha) = 1) > 0$, we prove a central limit theorem for $N({X_1, \dots, X_n}; \alpha)$.

Maxima of Poisson-like variables and related triangular arrays

Abstract: It is known that maxima of independent Poisson variables cannot be normalized to converge to a nondegenerate limit distribution. On the other hand, the Normal distribution approximates the Poisson distribution for large values of the Poisson mean, and maxima of random samples of Normal variables may be linearly scaled to converge to a classical extreme value distribution. We here explore the boundary between these two kinds of behavior. Motivation comes from the wish to construct models for the statistical analysis of extremes of background gamma radiation over the United Kingdom. The methods extend to row-wise maxima of certain triangular arrays, for which limiting distributions are also derived.

The waiting time distribution for the random order service $M/M/1$ queue

Abstract: The $M/M/1$ queue is considered in the case in which customers are served in random order. A formula is obtained for the distribution of the waiting time w in the stationary state. The formula is used to show that $P9w > t) \sim \alpha t^{-5/6} \exp (-\beta t - \gamma t^{1/3})$ as $t \to \infty$, with the constants $\alpha, \beta$, and $\gamma$ expressed as functions of the traffic intensity $\rho$. The distribution of w for the random order discipline is compared to that of the first in, first out discipline.

A scaling limit for queues in series

Abstract: We derive a law of large numbers for a tagged particle in the one-dimensional totally asymmetric simple exclusion process under a scaling different from the usual Euler scaling. By interpreting the particles as the servers of a series of queues we use this result to verify an open conjecture about the scaling behavior of the departure times from a long series of queues.

Functional large deviation principles for first-passage-time processes

Abstract: We apply an extended contraction principle and superexponential convergence in probability to show that a functional large deviation principle for a sequence of stochastic processes implies a corresponding functional large deviation principle for an associated sequence of first-passage-time or inverse processes. Large deviation principles are established for both inverse processes and centered inverse processes, based on corresponding results for the original process. We apply these results to obtain functional large deviation principles for renewal processes and superpositions of independent renewal processes.

Coexistence results for some competition models

Abstract: .yellow dwarf BYD is a serious widespread disease of small grains and grasses caused by a group of aphid transmitted viruses. Its symptoms are . chlorosis i.e., yellowing of plant tissue and stunting of the affected plant. BYD is an important agricultural disease since its affects large numbers of different grains throughout the world. The total yield loss in the United States is around 1 to 3 percent each year, but under favorable conditions, losses of 40% are not uncommon. The disease was first reported in the United . States by Galloway and Southwood 1890 but only much later recognized by . Oswald and Houston 1951 as being caused by a virus. . The barley yellow dwarf virus BYDV is a member of the luteoviruses. This group includes bean leaf roll, beet western yellows, carrot red leaf and potato leaf roll. These viruses are transmitted by aphids the ‘‘vector’’ for the . w disease and they are typically very host-specific see, e.g., Duffus, Falk, and .x Johnstone 1987 .

Explicit distributional results in pattern formation

Abstract: A new and unified approach is presented for constructing joint generating functions for quantities of interest associated with pattern formation in binary sequences. The method is based on results on exponential families of Markov chains. The tools we use are not only new for this area; they seem to be the right approach for deriving general explicit distributional results.

Optimization via trunk reservation in single resource loss systems under heavy traffic

Abstract: Trunk reservation is a simple, robust and extremely effective mechanism for controlling loss systems which allows priority to be given to chosen traffic streams. We consider the control of a single resource under a limiting regime in which capacity and arrival rates increase together. We obtain trunk reservation control policies which are asymptotically optimal when calls have differing capacity requirements, holding times, arrival rates and reward rates. The priority levels associated with these trunk reservation policies arise from an attainable bound on the performance of any control policy.

The quasi-stationary behavior of quasi-birth-and-death processes

Abstract: For evanescent Markov processes with a single transient communicating class, it is often of interest to examine the limiting probabilities that the process resides in the various transient states, conditional on absorption not having taken place. Such distributions are known as quasi-stationary (or limiting-conditional) distributions. In this paper we consider the determination of the quasi-stationary distribution of a general level-independent quasi-birth-and-death process (QBD). This distribution is shown to have a form analogous to the matrix-geometric form possessed by the stationary distribution of a positive recurrent QBD. We provide an algorithm for the explicit computation of the quasi-stationary distribution.

Increasing sequences of independent points on the planar lattice

Abstract: nŽ 2 number of points on an increasing path of these points in the square 0, n . Superadditivity and simple moment bounds imply that for some constant c, 1 lim L c a.s. n n n What is the exact value of c? The original problem concerned the longest increasing subsequence of a random permutation. This reformulation in terms of a planar Poisson point process is due to Hammersley 6 . In 1977 Vershik and Kerov derived the answer c 21 3. In the same year, Logan and Shepp independently showed that c 28 . Both proofs are combinatorial and make use of Young diagrams. Recently two proofs have Ž . appeared Aldous and Diaconis 1 and Seppalainen 12 that proceed by ¨¨ embedding the increasing sequences of points in an interacting particle system. In this paper we use the approach of 12 to solve the analogous problem on Ž.

On the convergence of multitype branching processes with varying environments

Abstract: Using the ergodic theory of nonnegative matrices, conditions are obtained for the L2 and almost sure convergence of a supercritical multitype branching process with varying environment, normed by its mean. We also give conditions for the extinction probability of the limit to equal that of the process. The theory developed allows for different types to grow at different rates, and an example of this is given, taken from the construction of a spatially inhomogeneous diffusion on the Sierpinski gasket.

Central limit theory for the number of seeds in a growth model in Rd with inhomogeneous poisson arrivals

Abstract: A Poisson point process $\Psi$ in d-dimensional Euclidean space and in time is used to generate a birth-growth model: seeds are born randomly at locations $x_i$ in $\mathbb{R}^d$ at times $t_i \epsilon [0, \infty)$. Once a seed is born, it begins to create a cell by growing radially in all directions with speed $v > 0$. Points of $\Psi$ contained in such cells are discarded, that is, thinned.We study the asymptotic distribution of the number of seeds in a region, as the volume of the region tends to infinity. When $d = 1$, we establish conditions under which the evolution over time of the number of seeds in a region is approximated by a Wiener process. When $d \geq 1$, we give conditions for asymptotic normality. Rates of convergence are given in all cases.

Average performance of a class of adaptive algorithms for global optimization

Abstract: We describe a class of adaptive algorithms for approximating the global minimum of a continuous function on the unit interval. The limiting distribution of the error is derived under the assumption of Wiener measure on the objective functions. For any > 0, we construct an algorithm which has error converging to zero at rate n ?(1?) in the number of function evaluations n. This convergence rate contrasts with the n ?1=2 rate of previously studied non-adaptive methods.

The large-scale structure of the universe and quasi-Voronoi tessellation of shock fronts in forced Burgers turbulence in $\bold R\sp {d}$

Abstract: Burgers turbulence is an accepted formalism for the adhesion model of the large-scale distribution of matter in the universe. The paper uses variational methods to establish evolution of quasi-Voronoi (curved boundaries) tessellation structure of shock fronts for solutions of the inviscid nonhomogeneous Burgers equation in $R^d$ in the presence of random forcing due to a degenerate potential. The mean rate of growth of the quasi-Voronoi cells is calculated and a scaled limit random tessellation structure is found. Time evolution of the probability that a cell contains a ball of a given radius is also determined.

Homogenization for time-dependent two-dimensional incompressible Gaussian flows

Abstract: We consider the diffusive scaling limit for the transport of a passive scalar in a two-dimensional time-dependent incompressible Gaussian velocity field and in the presence of molecular diffusivity. We prove that homogenization holds in this limiting regime and we derive some simple properties of the effective diffusivity tensor.

Predator-prey and host-parasite spatial stochastic models

Abstract: We consider two interacting particle systems on $\mathbf{Z}^d$ to model predator-prey and host-parasite interactions. In both models we have two types of $\mathbf{Z}^d$ particles (1 and 2) and each site in can be in one of four states: empty, occupied by a type 1 particle, occupied by a type 2 particle or occupied by two particles (one of each type). Each type gives birth to particles of the same type on nearest neighbor sites. The interaction between the two types of particles occurs only when a site is occupied by one particle of each type. For both models we show that coexistence and noncoexistence are possible in any dimension.

Motion in a Gaussian incompressible flow

Abstract: We prove that the solution of a system of random ordinary differential equations $d\mathbf{X}(t)/dt = \mathbf{V}(t, \mathbf{X}(t))$ with diffusive scaling, $\mathbf{X}_{\varepsilon}(t) = \varepsilon \mathbf{X}(t/ \varepsilon^2)$, converges weakly to a Brownian motion when $\varepsilon \downarrow 0$ We assume that $\mathbf{V}(t, \mathbf{x}), t \epsilon R, \mathbf{x} \epsilon R^d$ is a d-dimensional, random, incompressible, stationary Gaussian field which has mean zero and decorrelates in finite time

On using the first difference in the Stein-Chen method

Abstract: This paper investigates an alternative way of using the Stein-Chen method in Poisson approximations. There are three principal bounds stated in terms of reduced Palm probabilities for general point processes. The first two are for the accuracy of Poisson random variable approximation to the distribution of the number of points in a point process with respect to the total variation metric and the Wasserstein metric, and the third is for bounding the errors of Poisson process approximation to the distribution of a point process on a general compact space with respect to a Wasserstein metric. The bounds are frequently sharper than the previous results using the Stein-Chen method when the expected number of points is large.

About the multidimensional competitive learning vector quantization algorithm with constant gain

Abstract: The competitive learning vector quantization (CLVQ) algorithm with constant step $\varepsilon > 0$--also known as the Kohonen algorithm with 0 neighbors--is studied when the stimuli are i.i.d. vectors. Its first noticeable feature is that, unlike the one-dimensional case which has $n!$ absorbing subsets, the CLVQ algorithm is "irreducible on open sets" whenever the stimuli distribution has a path-connected support with a nonempty interior. Then the Doeblin recurrence (or uniform ergodicity) of the algorithm is established under some convexity assumption on the support. Several properties of the invariant probability measure $ u^{\varepsilon}$ are studied, including support location and absolute continuity with respect to the Lebesgue measure. Finally, the weak limit set of $ u^{\varepsilon}$ as $\varepsilon \to 0$ is investigated.

An extension to the renewal theorem and an application to risk theory

Abstract: In applied probability one is often interested in the asymptotic behavior of a certain quantity. If a regenerative phenomenon can be imbedded, then one has the problem that the event of interest may have occurred but cannot be observed at the renewal points. In this paper an extension to the renewal theorem is proved which shows that the quantity of interest converges. As an illustration an open problem in risk theory is solved.