Showing papers in "Journal of Applied Probability in 1998"
TL;DR: In this paper, it was shown that the spanning tree of time-minimizing paths from the origin in first passage percolation with exponential passage times has at least two topological ends with positive probability.
Abstract: An interacting particle system modelling competing growth on the Z 2 lattice is defined as follows. Each x E Z 2 is in one of the states {0, 1, 2). l's and 2's remain in their states for ever, while a 0 flips to a 1 (a 2) at a rate equal to the number of its neighbours which are in state 1 (2). This is a generalization of the well-known Richardson model. l's and 2's may be thought of as two types of infection, and 0's as uninfected sites. We prove that if we start with a single site in state 1 and a single site in state 2, then there is positive probability for the event that both types of infection reach infinitely many sites. This result implies that the spanning tree of time-minimizing paths from the origin in first passage percolation with exponential passage times has at least two topological ends with positive probability.
125 citations
TL;DR: In this article, a quantile dispersion measure is proposed to characterize different classes of ageing distributions. But it is weaker than the well known dispersive ordering and it retains most of its interesting properties.
Abstract: In this paper we introduce a quantile dispersion measure. We use it to characterize different classes of ageing distributions. Based on the quantile dispersion measure, we propose a new partial ordering for comparing the spread or dispersion in two probability distributions. This new partial ordering is weaker than the well known dispersive ordering and it retains most of its interesting properties.
114 citations
TL;DR: In this article, the authors studied the properties of exponential functionals of the linear Brownian motion in relation with the problem of a particle coupled to a heat bath in a Wiener potential.
Abstract: The paper deals with exponential functionals of the linear Brownian motion which arise in different contexts, such as continuous time finance models and one-dimensional disordered models. We study some properties of these exponential functionals in relation with the problem of a particle coupled to a heat bath in a Wiener potential. Explicit expressions for the distribution of the free energy are presented.
109 citations
TL;DR: In this paper, the authors introduced a population contact structure and derived the final size equation for a quite general superimposed epidemic process, characterized by the following two properties: (i) each individual contacts exactly k other individuals; (ii) these k acquaintances are a random sample of the (infinite) population.
Abstract: We introduce a certain population contact structure and derive, in three different ways, the final size equation for a quite general superimposed epidemic process. The contact structure is characterized by the following two properties: (i) each individual contacts exactly k other individuals; (ii) these k acquaintances are a random sample of the (infinite) population.
101 citations
TL;DR: In this paper, the authors considered a general SIR epidemic model with a critical scaling λ ≈ 1 + a/n 1/3, where n is the initial number of susceptibles and m is the number of infected.
Abstract: The distribution of the final size, K, in a general SIR epidemic model is considered in a situation when the critical parameter λ is close to 1. It is shown that with a ‘critical scaling’ λ ≈ 1 + a / n 1/3, m ≈ bn 1/3, where n is the initial number of susceptibles and m is the initial number of infected, then K / n 2/3 has a limit distribution when n → ∞. It can be described as that of T, the first passage time of a Wiener process to a parabolic barrier b + at − t 2/2. The proof is based on a diffusion approximation. Moreover, it is shown that the distribution of T can be expressed analytically in terms of Airy functions using the spectral representation connected with Airy's differential equation.
97 citations
TL;DR: In this paper, the authors show that the autocorrelation function of a class of processes constructed by embedding a Markov chain into a subexponential renewal process has a sub-exponential tail.
Abstract: Let {(Xn, Jn )} be a stationary Markov-modulated random walk on × E (E is finite), defined by its probability transition matrix measure F ={ F ij }, F ij (B) = [X1 ∈ B, J1 = j | J0 = i], B ∈ ( ), i, j ∈ E .I fF ij ([x, ∞))/(1 − H (x)) → W ij ∈ [0, ∞) ,a sx →∞ , for some long-tailed distribution function H , then the ascending ladder heights matrix distribution G+(x) (right Wiener–Hopf factor) has long-tailed asymptotics. If Xn 0, and H (x) is a subexponential distribution function, then the asymptotic behavior of the supremum of this random walk is the same as in the i.i.d. case, and it is given by [supn≥0 Sn > x ]→ (− Xn ) −1 ∞ x [Xn > u] du as x →∞ ,w hereSn = n Xk , S0 = 0. Two general queueing applications of this result are given. First, if the same asymptotic conditions are imposed on a Markov–modulated G/G/1 queue, then the waiting time distribution has the same asymptotics as the waiting time distribution of a GI / GI /1 queue, i.e., it is given by the integrated tail of the service time distribution function divided by the negative drift of the queue increment process. Second, the autocorrelation function of a class of processes constructed by embedding a Markov chain into a subexponential renewal process, has a subexponential tail. When a fluid flow queue is fed by these processes, the queue-length distribution is asymptotically proportional to its autocorrelation function.
91 citations
TL;DR: In this paper, the authors derived the asymptotic distribution for the extinction time as the population grows to infinity, under different initial conditions and for different values of the infection rate.
Abstract: The time until extinction for the closed SIS stochastic logistic epidemic model is investigated. We derive the asymptotic distribution for the extinction time as the population grows to infinity, under different initial conditions and for different values of the infection rate.
78 citations
TL;DR: In this article, the authors consider the question of which convergence properties of Markov chains are preserved under small perturbations, such as roundoff error from computer simulation and geometric ergodicity and rates of convergence.
Abstract: In this paper, we consider the question of which convergence properties of Markov chains are preserved under small perturbations. Properties considered include geometric ergodicity and rates of convergence. Perturbations considered include roundoff error from computer simulation. We are motivated primarily by interest in Markov chain Monte Carlo algorithms.
71 citations
TL;DR: In this article, a general convergence-to-the-coalescent theorem is presented, which works not only for a larger class of exchangeable models but also for a large class of non-exchangeable population models.
Abstract: A variety of convergence results for genealogical and line-of-descendent processes are known for exchangeable neutral population genetics models. A general convergence-to-the-coalescent theorem is presented, which works not only for a larger class of exchangeable models but also for a large class of non-exchangeable population models. The coalescence probability, i.e. the probability that two genes, chosen randomly without replacement, have a common ancestor one generation backwards in time, is the central quantity to analyse the ancestral structure.
67 citations
TL;DR: In this article, it was shown that all finite perturbations of stochastically monotone chains can be dominated by geometrically ergodic and irreducible chains.
Abstract: Let P be the transition matrix of a positive recurrent Markov chain on the integers, with invariant distribution π. If (n) P denotes the n x n ‘northwest truncation’ of P, it is known that approximations to π(j)/π(0) can be constructed from (n) P, but these are known to converge to the probability distribution itself in special cases only. We show that such convergence always occurs for three further general classes of chains, geometrically ergodic chains, stochastically monotone chains, and those dominated by stochastically monotone chains. We show that all ‘finite’ perturbations of stochastically monotone chains can be considered to be dominated by such chains, and thus the results hold for a much wider class than is first apparent. In the cases of uniformly ergodic chains, and chains dominated by irreducible stochastically monotone chains, we find practical bounds on the accuracy of the approximations.
59 citations
TL;DR: In this article, the authors consider the convex ordering for random vectors and some weaker versions of it, like convex orderings for linear combinations of random variables, and establish conditions of stochastic equality for the random vectors that are ordered by one of these orderings.
Abstract: We consider the convex ordering for random vectors and some weaker versions of it, like the convex ordering for linear combinations of random variables. First we establish conditions of stochastic equality for random vectors that are ordered by one of the convex orderings. Then we establish necessary and sufficient conditions for the convex ordering to hold in the case of multivariate normal distributions and sufficient conditions for the positive linear convex ordering (without the restriction to multi-normality).
TL;DR: In this article, it was shown that the only gain (average) optimal stationary policies with gain and bias which satisfy the optimality equation are of control limit type, that there are at most two and, if there are two, they occur consecutively.
Abstract: This paper studies an admission control M/M/1 queueing system. It shows that the only gain (average) optimal stationary policies with gain and bias which satisfy the optimality equation are of control limit type, that there are at most two and, if there are two, they occur consecutively. Conditions are provided which ensure the existence of two gain optimal control limit policies and are illustrated with an example. The main result is that bias optimality distinguishes these two gain optimal policies and that the larger of the two control limits is the unique bias optimal stationary policy. Consequently it is also Blackwell optimal. This result is established by appealing to the third optimality equation of the Markov decision process and some observations concerning the structure of solutions of the second optimality equation.
TL;DR: In this article, the authors consider a single server queue where the arrival process is a Markov-modulated Poisson process and service times are independent and identically distributed and independent from arrivals.
Abstract: We consider a single server queue where the arrival process is a Markov-modulated Poisson process and service times are independent and identically distributed and independent from arrivals. The underlying intensity process is assumed ergodic with generator cQ, c > 0. We prove under some monotonicity assumptions on Q that the stationary workload W(c) is decreasing in c with respect to the increasing convex ordering.
TL;DR: In this article, a class of stationary infinite-order moving average processes with margins in the class of infinitely divisible exponential dispersion models is considered, and the processes are constructed by means of the thinning operation of Joe (1996), generalizing the binomial thinning used by McKenzie (1986, 1988) and Al-Osh and Alzaid (1987) for integer-valued time series.
Abstract: We consider a class of stationary infinite-order moving average processes with margins in the class of infinitely divisible exponential dispersion models. The processes are constructed by means of the thinning operation of Joe (1996), generalizing the binomial thinning used by McKenzie (1986, 1988) and Al-Osh and Alzaid (1987) for integer-valued time series. As a special case we obtain a class of autoregressive moving average processes that are different from the ARMA models proposed by Joe (1996). The range of possible marginal distributions for the new models is extensive and includes all infinitely divisible distributions with finite moment generating functions, hereunder many known discrete, continuous and mixed distributions.
TL;DR: In this paper, Liu and Biggins and Kyprianou (1998) strengthened some results of Liu et al. concerning solutions to a certain functional equation associated with the branching random walk, and emphasized their importance in the context of travelling wave solutions to the KPP equation.
Abstract: In this short communication, some of the recent results of Liu (1998) and Biggins and Kyprianou (1997), concerning solutions to a certain functional equation associated with the branching random walk, are strengthened. Their importance is emphasized in the context of travelling wave solutions to a discrete version of the KPP equation and the connection with the behaviour of the rightmost particle in the nth generation.
TL;DR: In this paper, an explicit formula for the payoff and optimal stopping strategy of the optimal stopping problem with moving boundary is given, where the payoff is shown to be finite, if and only if μ 0 | | X t = g * (max 0≤t≤s X s )} where s ↦ g *(s) is the maximal solution of the (nonlinear) differential equation under the condition 0 0 X t ) = 1 − (σ 2 / 2 μ) and is sharp.
Abstract: Explicit formulas are found for the payoff and the optimal stopping strategy of the optimal stopping problem supτ E (max0≤t≤τ X t − c τ), where X = (X t ) t≥0 is geometric Brownian motion with drift μ and volatility σ > 0, and the supremum is taken over all stopping times for X. The payoff is shown to be finite, if and only if μ 0 | X t = g * (max0≤t≤s X s )} where s ↦ g *(s) is the maximal solution of the (nonlinear) differential equation under the condition 0 0 X t ) = 1 − (σ 2 / 2 μ) and is shown to be sharp. The method of proof relies upon a smooth pasting guess (for the Stephan problem with moving boundary) and the Ito–Tanaka formula (being applied two-dimensionally). The key point and main novelty in our approach is the maximality principle for the moving boundary (the optimal stopping boundary is the maximal solution of the differential equation obtained by a smooth pasting guess). We think that this principle is by itself of theoretical and practical interest.
TL;DR: In this article, the authors show that at each failure moment a repair can be performed with a random degree of repair including (as special cases) perfect, minimal and imperfect repair models.
Abstract: Stochastic comparison results for replacement policies are shown in this paper using the formalism of point processes theory. At each failure moment a repair is allowed. It is performed with a random degree of repair including (as special cases) perfect, minimal and imperfect repair models. Results for such repairable systems with schemes of planned replacements are also shown. The results are obtained by coupling methods for point processes.
TL;DR: In this article, the authors correct the central limit theorem of Rudnicki et al. and remove an error in its proof, based on an approximation by a martingale difference scheme.
Abstract: If ( F n ) n ∈ℕ is a sequence of independent and identically distributed random mappings from a second countable locally compact state space 𝕏 to 𝕏 which itself is independent of the 𝕏-valued initial variable X 0 , the discrete-time stochastic process ( X n ) n ≥0 , defined by the recursion equation X n = F n ( X n −1 ) for n ∈ℕ, has the Markov property. Since 𝕏 is Polish in particular, a complete metric d exists. The random mappings ( F n ) n ∈ℕ are assumed to satisfy ℙ-a.s.
Conditions on the distribution of l ( F n ) are given for the existence of an invariant distribution of X 0 making the process ( X n ) n ≥0 stationary and ergodic. Our main result corrects a central limit theorem by Łoskot and Rudnicki (1995) and removes an error in its proof. Instead of trying to compare the sequence φ ( X n ) n ≥0 for some φ : 𝕏 → ℝ with a triangular scheme of independent random variables our proof is based on an approximation by a martingale difference scheme.
TL;DR: In this article, the authors studied the queueing process in a class of D-policy models with Poisson bulk input, general service time, and four different vacation scenarios, among them a multiple vacation, single vacation and idle server.
Abstract: This paper studies the queueing process in a class of D-policy models with Poisson bulk input, general service time, and four different vacation scenarios, among them a multiple vacation, single vacation and idle server. The D-policy specifies a busy period discipline, which requires an idle or vacationing server to resume his service when the workload process crosses some fixed level D. The analysis of the queueing process is based on the theory of fluctuations for three-dimensional marked counting processes presented in the paper. For all models, we derive the stationary distributions for the embedded and continuous time parameter queueing processes in closed analytic forms and illustrate the results by a number of examples and applications.
TL;DR: The x log x condition is a fundamental criterion for the rate of growth of a general branching process, being equivalent to non-degeneracy of the limiting random variable as discussed by the authors.
Abstract: The x log x condition is a fundamental criterion for the rate of growth of a general branching process, being equivalent to non-degeneracy of the limiting random variable. In this paper we adopt the ideas from Lyons, Pemantle and Peres (1995) to present a new proof of this well-known theorem. The idea is to compare the ordinary branching measure on the space of population trees with another measure, the size-biased measure.
TL;DR: A new model, named controlled Markov set-chains, based on Markovset-chains is introduced, and its optimization under some partial order is discussed, to explain the theoretical results and the computation.
Abstract: In the framework of discounted Markov decision processes, we consider the case that the transition probability varies in some given domain at each time and its variation is unknown or unobservable. To this end we introduce a new model, named controlled Markov set-chains, based on Markov set-chains, and discuss its optimization under some partial order. Also, a numerical example is given to explain the theoretical results and the computation.
TL;DR: It turns out that whether or not homogenizing the individuals or households will result in an increased spread of infection actually depends on the infectiousness of the disease.
Abstract: We first study an epidemic amongst a population consisting of individuals with the same infectivity but with varying susceptibilities to the disease. The asymptotic final epidemic size is compared with the corresponding size for a homogeneous population. Then we group a heterogeneous population into households, assuming very high infectivity within households, and investigate how the global infection pressure is affected by rearranging individuals between the households. In both situations considered, it turns out that whether or not homogenizing the individuals or households will result in an increased spread of infection actually depends on the infectiousness of the disease.
TL;DR: In this paper, the stability of polling models is examined using associated fluid limit models, which generalize existing results in the literature or provide new stability conditions while in both cases providing simple and intuitive proofs of stability.
Abstract: The stability of polling models is examined using associated fluid limit models. Examples are presented which generalize existing results in the literature or provide new stability conditions while in both cases providing simple and intuitive proofs of stability. The analysis is performed for both general single server models and specific multiple server models.
TL;DR: Using large deviation principles, several mathematically founded linearization methods are proposed that use relative entropy, or Kullback information, of two probability measures, and Donsker-Varadhan entropy of a Gaussian measure relatively to a Markov kernel.
Abstract: Very little is known about the quantitative behaviour of dynamical systems with random excitation, unless the system is linear. Known techniques imply the resolution of parabolic partial differential equations (Fokker-Planck-Kolmogorov equation), which are degenerate and of high dimension and for which there is no effective known method of resolution. Therefore, users (physicists, mechanical engineers) concerned with such systems have had to design global linearization techniques, known as equivalent statistical linearization (Roberts and Spanos [5]). So far, there has been no rigorous justification of these techniques, with the notable exception of the paper by Frank Kozin [3]. In this contribution, using large deviation principles, several mathematically founded linearization methods are proposed. These principles use relative entropy, or Kullback information, of two probability measures, and Donsker-Varadhan entropy of a Gaussian measure relatively to a Markov kernel. The method of 'true linearization' ([5]) is justified.
TL;DR: In this article, bounds for the number of comparisons needed by Hoare's randomized selection algorithm FIND were obtained based on the construction and analysis of a suitable associated Markov chain, leading to the statement that FIND will need with probability 0.9 about 4.72 x n comparisons to find the median of a set S with n elements and n large.
Abstract: We obtain bounds for the distribution of the number of comparisons needed by Hoare's randomized selection algorithm FIND and give a new proof for Grubel and Rosler's (1996) result on the convergence of this distribution. Our approach is based on the construction and analysis of a suitable associated Markov chain. Some numerical results for the quantiles of the limit distributions are included, leading for example to the statement that, for a set S with n elements and n large, FIND will need with probability 0.9 about 4.72 x n comparisons to find the median of S.
TL;DR: In this article, the finiteness of moment generating functions and uniform recurrence assumptions were removed by using a new approach to remove these assumptions and derive asymptotic expansions for expected first passage times of Markov random walks.
Abstract: Previous work in extending Wald's equations to Markov random walks involves finiteness of moment generating functions and uniform recurrence assumptions. By using a new approach, we can remove these assumptions. The results are applied to establish finiteness of moments of ladder variables and to derive asymptotic expansions for expected first passage times of Markov random walks. Wiener-Hopf factorizations for Markov random walks are also applied to analyse ladder variables and related first passage problems.
TL;DR: In this article, it is shown that the number of physical minimal repairs is stochastically larger than that of statistical minimal repairs for k out of n systems with similar components and some majorization results are given for two component parallel systems with exponential components.
Abstract: The 'minimal' repair of a system can take several forms. Statistical or black box minimal repair at failure time t is equivalent to replacing the system with another functioning one of the same age, but without knowledge of precisely what went wrong with the system. Its major attribute is its mathematical tractability. In physical minimal repair, at system failure time t, we minimally repair the 'component' which brought the system down at time t. The work of Arjas and Norros, Finkelstein, and Natvig is reviewed. The concept of a rate function for minimal repairs of the statistical and physical types are discussed and developed. It is shown that the number of physical minimal repairs is stochastically larger than the number of statistical minimal repairs for k out of n systems with similar components. Some majorization results are given for physical minimal repair for two component parallel systems with exponential components.
TL;DR: In this paper, a central limit theorem for conditionally centred random fields, under a moment condition and strict positivity of the empirical variance per observation, was proved for Markov random fields.
Abstract: We prove a central limit theorem for conditionally centred random fields, under a moment condition and strict positivity of the empirical variance per observation. We use a random normalization, which fits non-stationary situations. The theorem applies directly to Markov random fields, including the cases of phase transition and lack of stationarity. One consequence is the asymptotic normality of the maximum pseudo-likelihood estimator for Markov fields in complete generality.
TL;DR: In this article, a Monte Carlo particle system approach to solve discrete time and nonlinear filtering problems is proposed and the main result is a uniform convergence theorem for a particle approximation of the nonlinear filter equation.
Abstract: The filtering problem concerns the estimation of a stochastic process X from its noisy partial information Y. With the notable exception of the linear-Gaussian situation, general optimal filters have no finitely recursive solution. The aim of this work is the design of a Monte Carlo particle system approach to solve discrete time and nonlinear filtering problems. The main result is a uniform convergence theorem. We introduce a concept of regularity and we give a simple ergodic condition on the signal semigroup for the Monte Carlo particle filter to converge in law and uniformly with respect to time to the optimal filter, yielding what seems to be the first uniform convergence result for a particle approximation of the nonlinear filtering equation.
TL;DR: In this article, the inverse absorption distribution is shown to be a q-Pascal analogue of the Kemp and Kemp (1991) q-binomial distribution, and exact expressions for the first two factorial moments are derived using q-series transformations of its probability generating function.
Abstract: The inverse absorption distribution is shown to be a q-Pascal analogue of the Kemp and Kemp (1991) q-binomial distribution. The probabilities for the direct absorption distribution are obtained via the inverse absorption probabilities and exact expressions for its first two factorial moments are derived using q-series transformations of its probability generating function. Alternative models for the distribution are given.