Showing papers in "Journal of Applied Probability in 2006"
TL;DR: In this article, the authors consider an insurance portfolio situation in which there is possible dependence between the waiting time for a claim and its actual size and obtain explicit exponential estimates for infinite and finite-time ruin probabilities in the case of light-tailed claim sizes.
Abstract: We consider an insurance portfolio situation in which there is possible dependence between the waiting time for a claim and its actual size. By employing the underlying random walk structure we obtain explicit exponential estimates for infinite- and finite-time ruin probabilities in the case of light-tailed claim sizes. The results are illustrated in several examples, worked out for specific dependence structures.
147 citations
TL;DR: This paper considers a variation of Poisson percolation models in which a connection between two points depends not only on their Euclidean distance, but also on the positions of all other points of the point process.
Abstract: Continuum percolation models in which pairs of points of a two-dimensional Poisson point process are connected if they are within some range of each other have been extensively studied. This paper considers a variation in which a connection between two points depends not only on their Euclidean distance, but also on the positions of all other points of the point process. This model has been recently proposed to model interference in radio communications networks. Our main result shows that, despite the infinite-range dependencies, percolation occurs in the model when the density λ of the Poisson point process is greater than the critical density value λc of the independent model, provided that interference from other nodes can be sufficiently reduced (without vanishing).
131 citations
TL;DR: In this paper, a stable Levy process is killed when it leaves the positive half-line, conditioning it to stay positive, and conditioned it to hit 0 continuously, and the authors obtain three different, positive, self-similar Markov processes which illustrate the three classes described by Lamperti.
Abstract: By variously killing a stable Levy process when it leaves the positive half-line, conditioning it to stay positive, and conditioning it to hit 0 continuously, we obtain three different, positive, self-similar Markov processes which illustrate the three classes described by Lamperti (1972). For each of these processes, we explicitly compute the infinitesimal generator and from this deduce the characteristics of the underlying Levy process in the Lamperti representation. The proof of this result bears on the behaviour at time 0 of stable Levy processes before their first passage time across level 0, which we describe here. As an application, for a certain class of Levy processes we give the law of the minimum before an independent exponential time. This provides the explicit form of the spatial Wiener–Hopf factor at a particular point and the value of the ruin probability for this class of Levy processes.
83 citations
TL;DR: In this paper, conditions under which X (1:i) decreases in i in the hazard rate order are derived for general (not necessarily exchangeable) random vectors, and these results are applied to obtain the limiting behaviour of the hazard-rate function of the lifetimes of coherent systems in reliability theory.
Abstract: Let X = (X 1 , X 2 ,..., X n ) be an exchangeable random vector, and write X (1:i) = min{X 1 , X 2 ,..., X i }, 1 ≤ i ≤ n. In this paper we obtain conditions under which X (1:i) decreases in i in the hazard rate order. A result involving more general (that is, not necessarily exchangeable) random vectors is also derived. These results are applied to obtain the limiting behaviour of the hazard rate function of the lifetimes of various coherent systems in reliability theory. The notions of the Samaniego signatures and the minimal signatures of such systems are extensively used in the paper. An interesting relationship between these two signatures is obtained. The results are illustrated in a series of examples.
72 citations
TL;DR: In this paper, a periodically inspected system with hidden failures is analyzed, where the rate of wear is modulated by a continuous-time Markov chain and additional damage is induced by a Poisson shock process.
Abstract: We analyze a periodically inspected system with hidden failures in which the rate of wear is modulated by a continuous-time Markov chain and additional damage is induced by a Poisson shock process. We explicitly derive the system's lifetime distribution and mean time to failure, as well as the limiting average availability. The main results are illustrated in two numerical examples.
67 citations
TL;DR: In this article, none-exponential asymptotics for solutions of two specific defective renewal equations are obtained, including the special cases of asymPTotics for a compound geometric distribution and the convolution of a compound geometrical distribution with a distribution function.
Abstract: Nonexponential asymptotics for solutions of two specific defective renewal equations are obtained. These include the special cases of asymptotics for a compound geometric distribution and the convolution of a compound geometric distribution with a distribution function. As applications of these results, we study the asymptotic behavior of the demographic birth rate of females, the perpetual put option in mathematics of finance, and the renewal function for terminating renewal processes.
61 citations
TL;DR: A new class of models is proposed, the generalizedvirtual age models, which generalize Kijima's virtual age models to the case where both preventive and corrective maintenances are present and characterizes the dependency between the two types of maintenance.
Abstract: In this paper we present a general framework for the modelling of the process of corrective and condition-based preventive maintenance actions for complex repairable systems A new class of models is proposed, the generalized virtual age models On the one hand, these models generalize Kijima's virtual age models to the case where both preventive and corrective maintenances are present On the other hand, they generalize the usual competing risks models to imperfect maintenance actions which do not renew the system A generalized virtual age model is defined by both a sequence of effective ages which characterizes the effects of both types of maintenance according to a classical virtual age model, and a usual competing risks model which characterizes the dependency between the two types of maintenance Several particular cases of the general model are derived
56 citations
TL;DR: In this article, the authors considered a Markov decision process with Borel state and action spaces and provided two average optimality inequalities of opposing directions and gave conditions for the existence of solutions to them.
Abstract: In this paper we study discrete-time Markov decision processes with Borel state and action spaces. The criterion is to minimize average expected costs, and the costs may have neither upper nor lower bounds. We first provide two average optimality inequalities of opposing directions and give conditions for the existence of solutions to them. Then, using the two inequalities, we ensure the existence of an average optimal (deterministic) stationary policy under additional continuity-compactness assumptions. Our conditions are slightly weaker than those in the previous literature. Also, some new sufficient conditions for the existence of an average optimal stationary policy are imposed on the primitive data of the model. Moreover, our approach is slightly different from the well-known ‘optimality inequality approach’ widely used in Markov decision processes. Finally, we illustrate our results in two examples.
55 citations
TL;DR: In this article, the authors argue that the underlying key to finding short paths is scale invariance, and they introduce a random-connection model that is related to continuum percolation and prove the existence of a unique scale-free model, among a large class of models, that allows the construction, with high probability, of short paths between pairs of points separated by any distance.
Abstract: The small-world phenomenon, the principle that we are all linked by a short chain of intermediate acquaintances, has been investigated in mathematics and social sciences. It has been shown to be pervasive both in nature and in engineering systems like the World Wide Web. Work of Jon Kleinberg has shown that people, using only local information, are very effective at finding short paths in a network of social contacts. In this paper we argue that the underlying key to finding short paths is scale invariance. In order to appreciate scale invariance we suggest a continuum setting, since true scale invariance happens at all scales, something which cannot be observed in a discrete model. We introduce a random-connection model that is related to continuum percolation, and we prove the existence of a unique scale-free model, among a large class of models, that allows the construction, with high probability, of short paths between pairs of points separated by any distance.
54 citations
TL;DR: In this article, the dependence structures for bivariate extremal events are analyzed using particular types of copula and weak convergence results for copulas along the lines of the Pickands-Balkema-de Haan theorem are provided.
Abstract: Dependence structures for bivariate extremal events are analyzed using particular types of copula. Weak convergence results for copulas along the lines of the Pickands-Balkema-de Haan theorem provide limiting dependence structures for bivariate tail events. A characterization of these limiting copulas is also provided by means of invariance properties. The results obtained are applied to the credit risk area, where, for intensity-based default models, stress scenario dependence structures for widely traded products such as credit default swap baskets or first-to-default contract types are proposed.
53 citations
TL;DR: In this article, a simple set of sufficient conditions for weak convergence of discrete-time, discrete-state Galton-Watson branching processes with immigration to continuous-time continuous-state branching process with immigration is provided.
Abstract: We provide a simple set of sufficient conditions for the weak convergence of discrete-time, discrete-state Galton-Watson branching processes with immigration to continuous-time, continuous-state branching processes with immigration.
TL;DR: In this paper, a continuous-time model with stationary increments for asset price {Pt} is proposed, which allows for skewness of returns and resembles closely that of Madan, Carr, and Chang (1998).
Abstract: A continuous-time model with stationary increments for asset price {Pt} is an extension of the symmetric subordinator model of Heyde (1999), and allows for skewness of returns. In the setting of independent variance-gamma-distributed returns the model resembles closely that of Madan, Carr, and Chang (1998). A simple choice of parameters renders {e_rr Pt} a familiar martingale. We then specify the activity time process, {Tt}, for which [Tt ? t] is asymptotically self-similar and [xt}, with zt = Tt ? 7}_i, is gamma distributed. This results in a skew variance-gamma distribution for each log price increment (return) Xt and a model for {Xt} which incorporates long-range dependence in squared returns. Our approach mirrors that for the (symmetric) Student process model of Heyde and Leonenko (2005), to which the present work is intended as a complement and a sequel. One intention is to compare, partly on the basis of fitting to data, versions of the general model wherein the returns have either (symmetric) t-distributions or variance-gamma distributions.
TL;DR: In this paper, the Laplace transform of the distribution of the maximum for a Levy process with positive jumps of phase type is derived, and error estimates show that this iteration converges geometrically fast.
Abstract: In this paper, we present an iterative procedure to calculate explicitly the Laplace transform of the distribution of the maximum for a Levy process with positive jumps of phase type. We derive error estimates showing that this iteration converges geometrically fast. Subsequently, we determine the Laplace transform of the law of the upcrossing ladder process and give an explicit pathwise construction of this process.
TL;DR: In this paper, the authors show how new fluctuation identities and their associated asymptotics, given in Vigon (2002), Kluppelberg et al. (2004), and Doney and Kyprianou (2006), provide the basis for establishing, in an elementary way, asymptic overshoot and undershoot distribitions for a general class of Levy insurance risk processes.
Abstract: In this short note we show how new fluctuation identities and their associated asymptotics, given in Vigon (2002), Kluppelberg et al. (2004) and Doney and Kyprianou (2006), provide the basis for establishing, in an elementary way, asymptotic overshoot and undershoot distribitions for a general class of Levy insurance risk processes. The results bring the earlier conclusions of Asmussen and Kluppelberg (1996) for the Cramer-Lundberg process into greater generality.
TL;DR: In this article, the authors characterize the critical cache size below which the dependency of requests dominates the cache performance, assuming that requests are generated by a nearly completely decomposable Markov-modulated process.
Abstract: It was recently proved by Jelenkovic and Radovanovic (2004) that the least-recently-used (LRU) caching policy, in the presence of semi-Markov-modulated requests that have a generalized Zipf's law popularity distribution, is asymptotically insensitive to the correlation in the request process. However, since the previous result is asymptotic, it remains unclear how small the cache size can become while still retaining the preceding insensitivity property. In this paper, assuming that requests are generated by a nearly completely decomposable Markov-modulated process, we characterize the critical cache size below which the dependency of requests dominates the cache performance. This critical cache size is small relative to the dynamics of the modulating process, and in fact is sublinear with respect to the sojourn times of the modulated chain that determines the dependence structure.
TL;DR: In this article, a scaling law for the critical SIS stochastic epidemic is proposed, where at time 0 the population consists of √ N infected and N - √ n susceptible individuals, and when the time and the number currently infected are both scaled by √N, the resulting process converges to a diffusion process related to the Feller diffusion by a change of drift.
Abstract: We exhibit a scaling law for the critical SIS stochastic epidemic. If at time 0 the population consists of √N infected and N - √N susceptible individuals, then when the time and the number currently infected are both scaled by √N, the resulting process converges, as N → oo, to a diffusion process related to the Feller diffusion by a change of drift. As a consequence, the rescaled size of the epidemic has a limit law that coincides with that of a first passage time for the standard Ornstein-Uhlenbeck process. These results are the analogs for the SIS epidemic of results of Martin-Lof (1998) and Aldous (1997) for the simple SIR epidemic.
TL;DR: In this article, a family of long-range percolation models (G p ) p>0 on ℤ d that allow dependence between edges and have the following connectivity properties for p ∈ (1/d, ∞): (i) the degree distribution of vertices in G p has a power-law distribution; (ii) the graph distance between points x and y is bounded by a multiple of log pd logpd | x - y | with probability 1 - o(1); and (iii) an adversary can delete a relatively small number of
Abstract: We consider a family of long-range percolation models (G p ) p>0 on ℤ d that allow dependence between edges and have the following connectivity properties for p ∈ (1/d, ∞): (i) the degree distribution of vertices in G p has a power-law distribution; (ii) the graph distance between points x and y is bounded by a multiple of log pd log pd | x - y | with probability 1 - o(1); and (iii) an adversary can delete a relatively small number of nodes from G p (ℤ d ∩ [0, n] d ), resulting in two large, disconnected subgraphs.
TL;DR: In this paper, the authors considered a single-server queueing system where customers arrive according to a Poisson process and the service times of the customers are independent and follow a Coxian distribution of order r. The system is subject to costs per unit time for holding a customer in the system.
Abstract: We consider a single-server queueing system at which customers arrive according to a Poisson process. The service times of the customers are independent and follow a Coxian distribution of order r. The system is subject to costs per unit time for holding a customer in the system. We give a closed-form expression for the average cost and the corresponding value function. The result can be used to derive nearly optimal policies in controlled queueing systems in which the service times are not necessarily Markovian, by performing a single step of policy iteration. We illustrate this in the model where a controller has to route to several single-server queues. Numerical experiments show that the improved policy has a close-to-optimal value. © Applied Probability Trust 2006.
TL;DR: In this paper, the authors considered stochastic recursive equations of sum type and of max type, and developed some new contraction properties of minimal L s -metrics which allow them to establish general existence and uniqueness results for solutions without imposing any moment conditions.
Abstract: In this paper we consider stochastic recursive equations ofsum type, X = Σ K i=1 A i X i +b, and of max type, X D/= max{A i X i + b i : 1 ≤ i ≤ k}, where A i , b i , and b are random, (X i ) are independent, identically distributed copies of X, and D/'=' denotes equality in distribution. Equations of these types typically characterize limits in the probabilistic analysis of algorithms, in combinatorial optimization problems, and in many other problems having a recursive structure. We develop some new contraction properties of minimal L s -metrics which allow us to establish general existence and uniqueness results for solutions without imposing any moment conditions. As an application we obtain a one-to-one relationship between the set of solutions to the homogeneous equation and the set of solutions to the inhomogeneous equation, for sum- and max-type equations. We also give a stochastic interpretation of a recent transfer principle of Rosler from nonnegative solutions of sum type to those of max type, by means of random scaled Weibull distributions.
TL;DR: This work extends the result to more realistic models constructed from a Poisson point process wherein each point is connected to all its neighbours within some fixed radius, and possesses random shortcuts to more distant nodes as described above.
Abstract: In a recent paper, Kleinberg (2000) considered a small-world network model consisting of a d-dimensional lattice augmented with shortcuts. The probability of a shortcut being present between two points decays as a power, r −α , of the distance, r, between them. Kleinberg showed that greedy routeing is efficient if α = d and that there is no efficient decentralised routeing algorithm if α �= d. The results were extended to a continuum model by Franceschetti and Meester (2003). In our work, we extend the result to more realistic models constructed from a Poisson point process wherein each point is connected to all its neighbours within some fixed radius, and possesses random shortcuts to more distant nodes as described above.
TL;DR: In this article, the authors characterize the continuous majorization of integrable functions introduced by Hardy, Littlewood, and Polya in terms of the discrete majorisation of finite-dimensional vectors, introduced by the same authors.
Abstract: We characterize the (continuous) majorization of integrable functions introduced by Hardy, Littlewood, and Polya in terms of the (discrete) majorization of finite-dimensional vectors, introduced by the same authors. The most interesting version of this result is the characterization of the (increasing) convex order for integrable random variables in terms of majorization of vectors of expected order statistics. Such a result includes, as particular cases, previous results by Barlow and Proschan and by Alzaid and Proschan, and, in a sense, completes the picture of known results on order statistics. Applications to other stochastic orders are also briefly considered.
TL;DR: In this paper, the authors considered a decomposable Galton-Watson process with individuals of two types, 0 and 1, and studied the properties of the waiting time to escape.
Abstract: The mathematical model we consider here is a decomposable Galton-Watson process with individuals of two types, 0 and 1 Individuals of type 0 are supercritical and can only produce individuals of type 0, whereas individuals of type 1 are subcritical and can produce individuals of both types The aim of this paper is to study the properties of the waiting time to escape, ie the time it takes to produce a type-0 individual that escapes extinction when the process starts with a type-1 individual With a view towards applications, we provide examples of populations in biological and medical contexts that can be suitably modeled by such processes
TL;DR: In this article, the authors obtained an expression for the correlation of the maxima of two correlated Brownian motions, and showed that the correlation is independent of the number of correlated motions.
Abstract: We obtain an expression for the correlation of the maxima of two correlated Brownian motions.
TL;DR: In this article, it is shown that all the existence, uniqueness, and convergence results of the finite-state case hold when the set of states is a general Borel space, provided that the optimal value function is bounded below.
Abstract: This paper is concerned with the analysis of Markov decision processes in which a natural form of termination ensures that the expected future costs are bounded, at least under some policies. Whereas most previous analyses have restricted attention to the case where the set of states is finite, this paper analyses the case where the set of states is not necessarily finite or even countable. It is shown that all the existence, uniqueness, and convergence results of the finite-state case hold when the set of states is a general Borel space, provided we make the additional assumption that the optimal value function is bounded below. We give a sufficient condition for the optimal value function to be bounded below which holds, in particular, if the set of states is countable.
TL;DR: In this paper, the authors consider a Levy process with no negative jumps, reflected at a stochastic boundary that is a positive constant multiple of an age process associated with a Poisson process and show that the stability condition for this process is identical to the one for the case of reflection at the origin.
Abstract: We consider a Levy process with no negative jumps, reflected at a stochastic boundary that is a positive constant multiple of an age process associated with a Poisson process. We show that the stability condition for this process is identical to the one for the case of reflection at the origin. In particular, there exists a unique stationary distribution that is independent of initial conditions. We identify the Laplace-Stieltjes transform of the stationary distribution and observe that it satisfies a decomposition property. In fact, it is a sum of two independent random variables, one of which has the stationary distribution of the process reflected at the origin, and the other the stationary distribution of a certain clearing process. The latter is itself distributed as an infinite sum of independent random variables. Finally, we discuss the tail behavior of the stationary distribution and in particular observe that the second distribution in the decomposition always has a light tail.
TL;DR: In this article, the authors introduced the concepts of instantaneous reversibility and instantaneous entropy production rate for inhomogeneous Markov chains with denumerable state spaces, and proved that the rotation number is 0 when it is instantaneously reversible or periodically reversible.
Abstract: In this paper we introduce the concepts of instantaneous reversibility and instantaneous entropy production rate for inhomogeneous Markov chains with denumerable state spaces. The following statements are proved to be equivalent: the inhomogeneous Markov chain is instantaneously reversible; it is in detailed balance; its entropy production rate vanishes. In particular, for a time-periodic birth-death chain, which can be regarded as a simple version of a physical model (Brownian motors), we prove that its rotation number is 0 when it is instantaneously reversible or periodically reversible. Hence, in our model of Markov chains, the directed transport phenomenon of Brownian motors can occur only in nonequilibrium and irreversible systems.
TL;DR: In this paper, the authors studied polynomial and geometric ergodicity for M/G/l -type Markov chains and Markov processes and obtained the radius of convergence of the generating function of the first return probability, which is very important in obtaining explicit bounds on geometric (exponential) convergence rates.
Abstract: In this paper we study polynomial and geometric (exponential) ergodicity for M/G/l -type Markov chains and Markov processes. First, practical criteria for M/G/l-type Markov chains are obtained by analyzing the generating function of the first return probability to level 0. Then the corresponding criteria for M/G/l-type Markov processes are given, using their h -approximation chains. Our method yields the radius of convergence of the generating function of the first return probability, which is very important in obtaining explicit bounds on geometric (exponential) convergence rates. Our results are illustrated, in the final section, in some examples.
TL;DR: In this article, the authors considered a class of risk processes with delayed claims, and provided ruin probability estimates under heavy tail conditions on the claim size distribution under the assumption of a subexponential distribution.
Abstract: We consider a class of risk processes with delayed claims, and we provide ruin probability estimates under heavy tail conditions on the claim size distribution. Keywords : Extreme value theory; Poisson process; regular variation; renewal process; ruin probability; shot noise process; subexponential distribution.
TL;DR: In this article, the authors developed and applied methods using gambling teams and martingales to find formulae for the expected value and the generating function of the waiting time to observation of an element of a finite collection of patterns in a sequence generated by a two-state Markov chain of first or higher order.
Abstract: Methods using gambling teams and martingales are developed and applied to find formulae for the expected value and the generating function of the waiting time to observation of an element of a finite collection of patterns in a sequence generated by a two-state Markov chain of first, or higher, order.
TL;DR: In this article, the limit behavior of a stationary moving average process with random coefficients was analyzed using a point process analysis, and the results showed that the limiting behavior of the stationary MA(∞) with random coefficient is similar to that of a standard MA( ∞) process.
Abstract: We consider a stationary moving average process with random coefficients, , generated by an array, { C t , k , t ∈ Z , k ≥ 0}, of random variables and a heavy-tailed sequence, { Z t , t ∈ Z }. We analyze the limit behavior using a point process analysis. As applications of our results we compare the limiting behavior of the moving average process with random coefficients with that of a standard MA(∞) process.