# Showing papers in "Journal of Applied Probability in 2016"

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TL;DR: In this paper, sufficient conditions for the permanence and ergodicity of a stochastic predator-prey model with a Beddington-DeAngelis functional response were derived.

Abstract: In this paper we derive sufficient conditions for the permanence and ergodicity of a stochastic predator–prey model with a Beddington–DeAngelis functional response. The conditions obtained are in fact very close to the necessary conditions. Both nondegenerate and degenerate diffusions are considered. One of the distinctive features of our results is that they enable the characterization of the support of a unique invariant probability measure. It proves the convergence in total variation norm of the transition probability to the invariant measure. Comparisons to the existing literature and matters related to other stochastic predator–prey models are also given.

52 citations

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TL;DR: The so-called Gerber–Shiu distribution for a ruin model where at each time the surplus process goes negative, an independent exponential clock is started and if the clock rings before the surplus becomes positive again then the insurance company is ruined.

Abstract: Inspired by works of Landriault et al. [11, 12], we study the Gerber{Shiu distribution at Parisian ruin with exponential implementation delays for a spectrally negative Levy insurance risk process. To be more specific, we study the so-called Gerber{Shiu distribution for a ruin model where at each time the surplus process goes negative, an independent exponential clock is started. If the clock rings before the surplus becomes positive again then the insurance company is ruined. Our methodology uses excursion theory for spectrally negative Levy processes and relies on the theory of so-called scale functions. In particular, we extend recent results of Landriault et al. [11, 12].

50 citations

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TL;DR: A stochastic coupling construction is developed to obtain the diffusion limit of the queue process in the Halfin‒Whitt heavy-traffic regime, and it is established that it does not depend on the value of d, implying that assigning tasks to idle servers is sufficient for diffusion level optimality.

Abstract: We consider a system of N parallel queues with identical exponential service rates and a single dispatcher where tasks arrive as a Poisson process. When a task arrives, the dispatcher always assigns it to an idle server, if there is any, and to a server with the shortest queue among d randomly selected servers otherwise (1≤d≤N). This load balancing scheme subsumes the so-called join-the-idle queue policy (d=1) and the celebrated join-the-shortest queue policy (d=N) as two crucial special cases. We develop a stochastic coupling construction to obtain the diffusion limit of the queue process in the Halfin‒Whitt heavy-traffic regime, and establish that it does not depend on the value of d, implying that assigning tasks to idle servers is sufficient for diffusion level optimality.

47 citations

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TL;DR: It is proved that the joint distribution of in-degree and out-degree has jointly regularly varying tails.

Abstract: The research of the authors was supported by MURI ARO
Grant W911NF-12-10385 to Cornell University

44 citations

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TL;DR: The matching model is introduced and it is proved that the model may be stable if and only if the matching graph is nonbipartite.

Abstract: We introduce and study a new model that we call the matching model. Items arrive one by one in a buffer and depart from it as soon as possible but by pairs. The items of a departing pair are said to be matched. There is a finite set of classes 𝒱 for the items, and the allowed matchings depend on the classes, according to a matching graph on 𝒱. Upon arrival, an item may find several possible matches in the buffer. This indeterminacy is resolved by a matching policy. When the sequence of classes of the arriving items is independent and identically distributed, the sequence of buffer-content is a Markov chain, whose stability is investigated. In particular, we prove that the model may be stable if and only if the matching graph is nonbipartite.

44 citations

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TL;DR: In this article, the short-range dependence (SRD) property of the increments of the fractional Poisson process was discussed, and it was shown that fractional negative binomial process (FNBP) has the same property.

Abstract: We discuss the short-range dependence (SRD) property of the increments of the fractional Poisson process, called the fractional Poissonian noise. We also establish that the fractional negative binomial process (FNBP) has the long-range dependence (LRD) property, while the increments of the FNBP have the SRD property. Our definitions of the SRD/LRD properties are similar to those for a stationary process and different from those recently used in Biard and Saussereau (2014).

28 citations

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TL;DR: A limit law for normalized random means is proposed that exhibits such heavy tails irrespective of the distribution of the underlying sampling units: the limit is a t-distribution if the random variables have finite variances.

Abstract: One often observed empirical regularity is a power-law behavior of the tails of some distribution of interest. We propose a limit law for normalized random means that exhibits such heavy tails irrespective of the distribution of the underlying sampling units: the limit is a t-distribution if the random variables have finite variances. The generative scheme is then extended to encompass classic limit theorems for random sums. The resulting unifying framework has wide empirical applicability which we illustrate by considering two empirical regularities in two different fields. First, we turn to urban geography and explain why city-size growth rates are approximately t-distributed, using a model of random sector growth based on the central place theory. Second, turning to an issue in finance, we show that high-frequency stock index returns can be modeled as a generalized asymmetric Laplace process. These empirical illustrations elucidate the situations in which heavy tails can arise.

23 citations

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TL;DR: It is proved that under appropriate assumptions, the random-walk Metropolis algorithm in d dimensions takes O(d) iterations to converge to stationarity, while the Metropolis-adjusted Langevin algorithm takes O (d1/3) iterations.

Abstract: We connect known results about diffusion limits of Markov chain Monte Carlo (MCMC) algorithms to the computer science notion of algorithm complexity. Our main result states that any weak limit of a Markov process implies a corresponding complexity bound (in an appropriate metric). We then combine this result with previously-known MCMC diffusion limit results to prove that under appropriate assumptions, the random-walk Metropolis algorithm in d dimensions takes O(d) iterations to converge to stationarity, while the Metropolis-adjusted Langevin algorithm takes O(d1/3) iterations to converge to stationarity.

22 citations

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TL;DR: This paper connects Archimedean survival processes (ASPs) with the theory of Markov copulas, and presents some new properties of ASPs related to their dependency structure.

Abstract: In this paper we connect Archimedean survival processes (ASPs) with the theory of Markov copulas. ASPs were introduced by Hoyle and Menguturk (2013) to model the realized variance of two assets. We present some new properties of ASPs related to their dependency structure. We study weak and strong Markovian consistency properties of ASPs. An ASP is weak Markovian consistent, but generally not strong Markovian consistent. Our results contain necessary and sufficient conditions for an ASP to be strong Markovian consistent. These properties are closely related to the concept of Markov copulas, which is very useful in modelling different dependence phenomena. At the end we present possible applications.

18 citations

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TL;DR: This paper provides a comparison of the successive lumping (SL) methodology developed in Katehakis et al. (2015) with the popular lattice path counting (Mohanty (1979)) in obtaining rate matrices for queueing models, satisfying the specific quasi birth and death structure as in Van Leeuwaarden et al (2009).

Abstract: This paper provides a comparison of the successive lumping (SL) methodology developed in Katehakis et al. (2015) with the popular lattice path counting (Mohanty (1979)) in obtaining rate matrices for queueing models, satisfying the specific quasi birth and death structure as in Van Leeuwaarden et al. (2009) and Van Leeuwaarden and Winands (2006). The two methodologies are compared both in terms of applicability requirements and numerical complexity by analyzing their performance for the same classical queueing models considered in Van Leeuwaarden et al. (2009). The main findings are threefold. First, when both methods are applicable, the SL-based algorithms outperform the lattice path counting algorithm (LPCA). Second, there are important classes of problems (for example, models with (level) nonhomogenous rates or with finite state spaces) for which the SL methodology is applicable and for which the LPCA cannot be used. Third, another main advantage of SL algorithms over lattice path counting is that the former includes a method to compute the steady state distribution using this rate matrix.

17 citations

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TL;DR: Closed-form expressions for the distribution of the virtual (actual) queueing time for the BMAP/R/1 and BM AP/D/1 queues, where `R' represents a class of distributions having rational Laplace‒Stieltjes transforms, are presented.

Abstract: In this paper we present closed-form expressions for the distribution of the virtual (actual) queueing time for the BMAP/R/1 and BMAP/D/1 queues, where `R' represents a class of distributions having rational Laplace‒Stieltjes transforms. The closed-form analysis is based on the roots of the underlying characteristic equation. Numerical aspects have been tested for a variety of arrival and service-time distributions and results are matched with those obtained using the matrix-analytic method (MAM). Further, a comparative study of computation time of the proposed method with the MAM has been carried out. Finally, we also present closed-form expressions for the distribution of the virtual (actual) system time. The proposed method is analytically quite simple and easy to implement.

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TL;DR: An asymptotic expression is derived for the expected time to extinction of the SIS epidemic, but now starting at the endemic state, which is not independent of the distribution of Q.

Abstract: We study continuous-time birth–death type processes, where individuals have independent and identically distributed lifetimes, according to a random variable Q, with E[Q] = 1, and where the birth rate if the population is currently in state (has size) n is α(n). We focus on two important examples, namely α(n) = λn being a branching process, and α(n) = λn(N −n)/N which corresponds to an SIS (susceptible → infective → susceptible) epidemic model in a homogeneously mixing community of fixed size N. The processes are assumed to start with a single individual, i.e. in state 1. Let T , An, C, and S denote the (random) time to extinction, the total time spent in state n, the total number of individuals ever alive, and the sum of the lifetimes of all individuals in the birth–death process, respectively. We give expressions for the expectation of all these quantities and show that these expectations are insensitive to the distribution of Q. We also derive an asymptotic expression for the expected time to extinction of the SIS epidemic, but now starting at the endemic state, which is not independent of the distribution of Q. The results are also applied to the household SIS epidemic, showing that, in contrast to the household SIR (susceptible → infective → recovered) epidemic, its threshold parameter R∗ is insensitive to the distribution of Q.

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TL;DR: It is shown that, viewed at suitably chosen times increasing to ∞, the process converges to the fluid limit, governed by the same dynamics, but with a random initial condition.

Abstract: In the paper we present a phenomenon occurring in population processes that start near 0 and have large carrying capacity. By the classical result of Kurtz (1970), such processes, normalized by the carrying capacity, converge on finite intervals to the solutions of ordinary differential equations, also known as the fluid limit. When the initial population is small relative to the carrying capacity, this limit is trivial. Here we show that, viewed at suitably chosen times increasing to ∞, the process converges to the fluid limit, governed by the same dynamics, but with a random initial condition. This random initial condition is related to the martingale limit of an associated linear birth-and-death process.

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TL;DR: In this article, the problem of inferring the distribution of a random vector given that its norm is large requires modeling a homogeneous limiting density, and an approach based on graphical models which is suitable for high-dimensional vectors is proposed.

Abstract: The problem of inferring the distribution of a random vector given that its norm is large requires modeling a homogeneous limiting density. We suggest an approach based on graphical models which is suitable for high-dimensional vectors. We introduce the notion of one-component regular variation to describe a function that is regularly varying in its first component. We extend the representation and Karamata's theorem to one-component regularly varying functions, probability distributions and densities, and explain why these results are fundamental in multivariate extreme-value theory. We then generalize the Hammersley–Clifford theorem to relate asymptotic conditional independence to a factorization of the limiting density, and use it to model multivariate tails.

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TL;DR: A functional central limit theorem is proved for the tail process (W ∞(θ)-W n+r (θ)) r∈ℕ0 and a law of the iterated logarithm for W ∞-W n ( θ) as n→∞ is proved.

Abstract: Let (W n (θ)) n∈ℕ0 be the Biggins martingale associated with a supercritical branching random walk, and denote by W_∞(θ) its limit. Assuming essentially that the martingale (W n (2θ)) n∈ℕ0 is uniformly integrable and that var W 1(θ) is finite, we prove a functional central limit theorem for the tail process (W ∞(θ)-W n+r (θ)) r∈ℕ0 and a law of the iterated logarithm for W ∞(θ)-W n (θ) as n→∞.

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TL;DR: A representation is given for the distribution of the first jump time of the process with jump size in a given Borel set and the equivalence of this distribution and the total Lévy measure is studied.

Abstract: We study the distributional properties of jumps in a continuous-state branching process with immigration. In particular, a representation is given for the distribution of the first jump time of the process with jump size in a given Borel set. From this result we derive a characterization for the distribution of the local maximal jump of the process. The equivalence of this distribution and the total Levy measure is then studied. For the continuous-state branching process without immigration, we also study similar problems for its global maximal jump.

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TL;DR: It is shown that the urns synchronize almost surely and that the fraction of balls of each colour converges to the deterministic limit of one-half, which matches with the limit known for a single Friedman urn.

Abstract: We consider a model of N interacting two-colour Friedman urns. The interaction model considered is such that the reinforcement of each urn depends on the fraction of balls of a particular colour in that urn as well as the overall fraction of balls of that colour in all the urns combined together. We show that the urns synchronize almost surely and that the fraction of balls of each colour converges to the deterministic limit of one-half, which matches with the limit known for a single Friedman urn. Furthermore, we use the notion of stable convergence to obtain limit theorems for fluctuations around the synchronization limit.

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TL;DR: Sufficient conditions for the excess wealth order are provided based on properties of the quantile functions which are useful when the dispersive order does not hold and it is shown how these results can provide comparisons of quantities of interest in reliability and insurance.

Abstract: The purpose of this paper is twofold. On the one hand, we provide sufficient conditions for the excess wealth order. These conditions are based on properties of the quantile functions which are useful when the dispersive order does not hold. On the other hand, we study sufficient conditions for the comparison in the increasing convex order of spacings of generalized order statistics. These results will be combined to show how we can provide comparisons of quantities of interest in reliability and insurance.

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TL;DR: A model where individuals move between a community and their household during the course of the day, only infecting within their current group is developed, able to derive the probability of a major epidemic outbreak.

Abstract: During the course of a day an individual typically mixes with different groups of individuals. Epidemic models incorporating population structure with individuals being able to infect different groups of individuals have received extensive attention in the literature. However, almost exclusively the models assume that individuals are able to simultaneously infect members of all groups, whereas in reality individuals will typically only be able to infect members of any group they currently reside in. In the current work we develop a model where individuals move between a community and their household during the course of the day, only infecting within their current group. By defining a novel branching process approximation with an explicit expression for the probability generating function of the offspring distribution, we are able to derive the probability of a major epidemic outbreak.

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TL;DR: In this article, it was shown that the reflected AR(1) process converges to a reflected Ornstein-Uhlenbeck process under heavy traffic scaling, and the corresponding steady-state distribution converged to the distribution of a normal random variable conditioned on being positive.

Abstract: In this paper we study a reflected AR(1) process, i.e. a process (Z(n))(n) obeying the recursion Z(n+1) = max{aZ(n) + X-n, 0}, with (X-n)(n) a sequence of independent and identically distributed (i.i.d.) random variables. We find explicit results for the distribution of Z(n) (in terms of transforms) in case X-n can be written as Y-n - B-n, with (B-n) n being a sequence of independent random variables which are all Exp(lambda) distributed, and (Y-n)(n) i.i.d.; when vertical bar a vertical bar < 1 we can also perform the corresponding stationary analysis. Extensions are possible to the case that (B-n)(n) are of phase-type. Under a heavy-traffic scaling, it is shown that the process converges to a reflected Ornstein-Uhlenbeck process; the corresponding steady-state distribution converges to the distribution of a normal random variable conditioned on being positive.

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TL;DR: This work addresses two fundamental sensor allocation problems and investigates asymptotic behaviour of optimal costs as n increases to infinity, in which the overlaps between any two sensors can not exceed a certain parameter.

Abstract: A large number n of sensors (finite connected intervals) are placed randomly on the real line so that the distances between the consecutive midpoints are independent random variables with expectation inversely proportional to n. In this work we address two fundamental sensor allocation problems. The interference problem tries to reallocate the sensors from their initial positions to eliminate overlaps. The coverage problem, on the other hand, allows overlaps, but tries to eliminate uncovered spaces between the originally placed sensors. Both problems seek to minimize the total sensor movement while reaching their respective goals. Using tools from queueing theory, Skorokhod reflections, and weak convergence, we investigate asymptotic behaviour of optimal costs as n increases to ∞. The introduced methodology is then used to address a more complicated, modified coverage problem, in which the overlaps between any two sensors can not exceed a certain parameter.

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TL;DR: Under simple broad conditions, the ergodicity of such Markov chains is established and closed-form expressions for the stationary densities of the chains when the proportions are beta distributed with the first parameter equal to 1 are derived.

Abstract: We consider a class of discrete-time Markov chains with state space [0, 1] and the following dynamics. At each time step, first the direction of the next transition is chosen at random with probability depending on the current location. Then the length of the jump is chosen independently as a random proportion of the distance to the respective end point of the unit interval, the distributions of the proportions being fixed for each of the two directions. Chains of that kind were the subjects of a number of studies and are of interest for some applications. Under simple broad conditions, we establish the ergodicity of such Markov chains and then derive closed-form expressions for the stationary densities of the chains when the proportions are beta distributed with the first parameter equal to 1. Examples demonstrating the range of stationary distributions for processes described by this model are given, and an application to a robot coverage algorithm is discussed.

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TL;DR: The moderate deviation principle is established for a class of stochastic partial differential equations with non-Lipschitz continuous coefficients and derived for two important population models: super-Brownian motion and the Fleming–Viot process.

Abstract: We establish the moderate deviation principle for the solutions of a class of stochastic partial differential equations with non-Lipschitz continuous coefficients. As an application, we derive the moderate deviation principle for two important population models: super-Brownian motion and the Fleming-Viot process.

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TL;DR: A law of large numbers, central limit theorem, and large and moderate deviation principles for log ZNt are shown.

Abstract: Let {Zn, n = 0, 1, 2, . . .} be a supercritical branching process, {Nt, t ≥ 0} be a Poisson process independent of {Zn, n = 0, 1, 2, . . .}, then {ZNt, t ≥ 0} is a supercritical Poisson random indexed branching process. We show a law of large numbers, central limit theorem, and large and moderate deviation principles for log ZNt.

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TL;DR: In this article, the proportion of black balls at time n and 0≤L L is replaced together with a random number Rn of red balls, and under mild conditions it is shown that Zn→a.s.Z for some random variable Z, and Dn≔√n(Zn-Z)→

Abstract: An urn contains black and red balls. Let Zn be the proportion of black balls at time n and 0≤L L, then bn is replaced together with a random number Rn of red balls. Otherwise, no additional balls are added, and bn alone is replaced. In this paper we assume that Rn=Bn. Then, under mild conditions, it is shown that Zn→a.s.Z for some random variable Z, and Dn≔√n(Zn-Z)→

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TL;DR: This paper builds the increasing convex (concave) order for the scalar product of random vectors with an upper (lower) tail permutation decreasing joint density and revisit allocations of portfolio risks in financial engineering and of coverage limits and deductibles in insurance.

Abstract: In this paper we build the increasing convex (concave) order for the scalar product of random vectors with an upper (lower) tail permutation decreasing joint density. As applications, we revisit allocations of portfolio risks in financial engineering and of coverage limits and deductibles in insurance. Some related results in the literature are substantially updated.

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TL;DR: Some central limit theorems in the sense of stable convergence and of almost sure conditional convergence are proved, which are stronger than convergence in distribution.

Abstract: We consider a variant of the randomly reinforced urn where more balls can be simultaneously drawn out and balls of different colors can be simultaneously added. More precisely, at each time-step, the conditional distribution of the number of extracted balls of a certain color, given the past, is assumed to be hypergeometric. We prove some central limit theorems in the sense of stable convergence and of almost sure conditional convergence, which are stronger than convergence in distribution. The proven results provide asymptotic confidence intervals for the limit proportion, whose distribution is generally unknown. Moreover, we also consider the case of more urns subjected to some random common factors.

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TL;DR: In this paper, the authors used the general Crump-Mode-Jagers branching process to model an outbreak of an infectious disease under mild assumptions, and derived a formula for the limit of the frequency of the occurrences of a given shape in a general tree.

Abstract: The shapes of branching trees have been linked to disease transmission patterns. In this paper we use the general Crump‒Mode‒Jagers branching process to model an outbreak of an infectious disease under mild assumptions. Introducing a new class of characteristic functions, we are able to derive a formula for the limit of the frequency of the occurrences of a given shape in a general tree. The computational challenges concerning the evaluation of this formula are in part overcome using the jumping chronological contour process. We apply the formula to derive the limit of the frequency of cherries, pitchforks, and double cherries in the constant-rate birth‒death model, and the frequency of cherries under a nonconstant death rate.

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TL;DR: The longest stretch $L(n)$ of consecutive heads in $n$ i.i.d. coin tosses is seen from the prism of large deviations and precise asymptotics are established for the moment generating function of L(n).

Abstract: The longest stretch L(n) of consecutive heads in n independent and identically distributed coin tosses is seen from the prism of large deviations. We first establish precise asymptotics for the mom ...