Showing papers in "Journal of Applied Probability in 2017"
TL;DR: The large deviation principle for epidemic models obtained by applying the random time changes of Ethier and Kurtz (2005) to a collection of Poisson processes is shown and the approach followed by Dolgoarshinnykh in the case of the SIR epidemic model is generalised.
Abstract: We consider a general class of epidemic models obtained by applying the random time changes of Ethier and Kurtz (2005) to a collection of Poisson processes and we show the large deviation principle for such models. We generalise the approach followed by Dolgoarshinnykh (2009) in the case of the SIR epidemic model. Thanks to an additional assumption which is satisfied in many examples, we simplify the recent work of Kratz and Pardoux (2017).
31 citations
TL;DR: The results suggest that the CRE can be viewed as an alternative entropy (dispersion) measure to classical Shannon entropy.
Abstract: Recently, Rao et al. (2004) introduced an alternative measure of uncertainty known as the cumulative residual entropy (CRE). It is based on the survival (reliability) function F instead of the probability density function f used in classical Shannon entropy. In reliability based system design, the performance characteristics of the coherent systems are of great importance. Accordingly, in this paper, we study the CRE for coherent and mixed systems when the component lifetimes are identically distributed. Bounds for the CRE of the system lifetime are obtained. We use these results to propose a measure to study if a system is close to series and parallel systems of the same size. Our results suggest that the CRE can be viewed as an alternative entropy (dispersion) measure to classical Shannon entropy.
29 citations
TL;DR: In this article, the join-idle-queue routeing algorithm is studied, under which an arriving customer is sent to an idle server, if such is available, and to a randomly uniformly chosen server, otherwise.
Abstract: A parallel server system with n identical servers is considered. The service time distribution has a finite mean 1 / μ, but otherwise is arbitrary. Arriving customers are routed to one of the servers immediately upon arrival. The join-idle-queue routeing algorithm is studied, under which an arriving customer is sent to an idle server, if such is available, and to a randomly uniformly chosen server, otherwise. We consider the asymptotic regime where n → ∞ and the customer input flow rate is λn. Under the condition λ / μ < ½, we prove that, as n → ∞, the sequence of (appropriately scaled) stationary distributions concentrates at the natural equilibrium point, with the fraction of occupied servers being constant at λ / μ. In particular, this implies that the steady-state probability of an arriving customer waiting for service vanishes.
27 citations
TL;DR: This work proves nonuniversality results for first-passage percolation on the configuration model with independent and identically distributed (i.i.d.) degrees having infinite variance by focusing on the weight of the optimal path between two uniform vertices.
Abstract: We prove nonuniversality results for first-passage percolation on the configuration model with independent and identically distributed (i.i.d.) degrees having infinite variance. We focus on the weight of the optimal path between two uniform vertices. Depending on the properties of the weight distribution, we use an example-based approach and show that rather different behaviours are possible. When the weights are almost surely larger than a constant, the weight and number of edges in the graph grow proportionally to log log n, as for the graph distances. On the other hand, when the continuous-time branching process describing the first-passage percolation exploration through the graph reaches infinitely many vertices in finite time, the weight converges to the sum of two i.i.d. random variables representing the explosion times of the continuous-time processes started from the two sources. This nonuniversality is in sharp contrast to the setting where the degree sequence has a finite variance, Bhamidi et al. (2012).
20 citations
TL;DR: A method by recursion is developed to calculate the cost components and the corresponding premium levels in this extended epidemic model of an extended SIR epidemic in which the removal and infection rates may depend on the number of registered removals.
Abstract: This paper aims to apply simple actuarial methods to build an insurance plan protecting against an epidemic risk in a population. The studied model is an extended SIR epidemic in which the removal and infection rates may depend on the number of registered removals. The costs due to the epidemic are measured through the expected epidemic size and infectivity time. The premiums received during the epidemic outbreak are measured through the expected susceptibility time. Using martingale arguments, a method by recursion is developed to calculate the cost components and the corresponding premium levels in this extended epidemic model. Some numerical examples illustrate the effect of removals and the premium calculation in an insurance plan.
19 citations
TL;DR: Comparisons of the failure times and interfailure times of two systems based on a replacement policy proposed by Kapodistria and Psarrakos (2012) are provided and it is shown that when the first failure times are ordered in terms of the dispersive order, then the successive interfailur times are orders in the usual stochastic order.
Abstract: We provide some results for the comparison of the failure times and interfailure times of two systems based on a replacement policy proposed by Kapodistria and Psarrakos (2012). In particular, we show that when the first failure times are ordered in terms of the dispersive order (or, the excess wealth order), then the successive interfailure times are ordered in terms of the usual stochastic order (respectively, the increasing convex order). As a consequence, we provide comparison results for the cumulative residual entropies of the systems and their dynamic versions.
18 citations
TL;DR: It is proved that the asymptotic distribution of the measure – with respect to dμ = f(x)dx – of the cell containing X 1 given X 1 = x is independent of x and the density f, and the distribution becomes more concentrated as d becomes large.
Abstract: $n$ independent random points drawn from a density $f$ in $R^d$ define a random Voronoi partition. We study the measure of a typical cell of the partition. We prove that the asymptotic distribution of the probability measure of the cell centered at a point $x \in R^d$ is independent of $x$ and the density $f$. We determine all moments of the asymptotic distribution and show that the distribution becomes more concentrated as $d$ becomes large. In particular, we show that the variance converges to zero exponentially fast in $d$. %We also study the measure of the largest cell of the partition. %{\red We also obtain a density-free bound for the rate of convergence of the diameter of a typical Voronoi cell.
17 citations
TL;DR: In this article, for spectrally negative Levy processes, the authors find joint Laplace transforms involving the last exit time (from a semi-infinite interval), the value of the process at the last departure time, and the associated occupation time.
Abstract: Using a new approach, for spectrally negative Levy processes we find joint Laplace transforms involving the last exit time (from a semiinfinite interval), the value of the process at the last exit time, and the associated occupation time, which generalize some previous results.
17 citations
TL;DR: In this paper, it was shown that the mixed preconditioned Crank-Nicolson (MpCN) algorithm has geometric ergodicity even for heavy-tailed target distributions.
Abstract: We describe the ergodic properties of some Metropolis–Hastings algorithms for heavy-tailed target distributions. The results of these algorithms are usually analyzed under a subgeometric ergodic framework, but we prove that the mixed preconditioned Crank–Nicolson (MpCN) algorithm has geometric ergodicity even for heavy-tailed target distributions. This useful property comes from the fact that, under a suitable transformation, the MpCN algorithm becomes a random-walk Metropolis algorithm.
15 citations
TL;DR: In this article, the authors study the joint law of the first passage time of the drawdown (respectively, drawup) process, its overshoot, and the maximum of the underlying process at this passage time.
Abstract: Drawdown (respectively, drawup) of a stochastic process, also referred as the reflected process at its supremum (respectively, infimum), has wide applications in many areas including financial risk management, actuarial mathematics, and statistics. In this paper, for general time-homogeneous Markov processes, we study the joint law of the first passage time of the drawdown (respectively, drawup) process, its overshoot, and the maximum of the underlying process at this first passage time. By using short-time pathwise analysis, under some mild regularity conditions, the joint law of the three drawdown quantities is shown to be the unique solution to an integral equation which is expressed in terms of fundamental two-sided exit quantities of the underlying process. Explicit forms for this joint law are found when the Markov process has only one-sided jumps or is a Levy process (possibly with two-sided jumps). The proposed methodology provides a unified approach to study various drawdown quantities for the general class of time-homogeneous Markov processes.
15 citations
TL;DR: In this paper, the authors consider the problem of optimal investment with intermediate consumption in a general semimartingale model of an incomplete market, with preferences being represented by a utility stochastic field.
Abstract: We consider the problem of optimal investment with intermediate consumption in a general semimartingale model of an incomplete market, with preferences being represented by a utility stochastic field. We show that the key conclusions of the utility maximization theory hold under the assumptions of no unbounded profit with bounded risk and of the finiteness of both primal and dual value functions. Copyright © 2017 Applied Probability Trust.
TL;DR: The unconstrained exponential family of random graphs assumes no prior knowledge of the graph before sampling, but it is natural to consider situations where partial information about the graph is known, for example the total number of edges as mentioned in this paper.
Abstract: The unconstrained exponential family of random graphs assumes no prior knowledge of the graph before sampling, but it is natural to consider situations where partial information about the graph is known, for example the total number of edges. What does a typical random graph look like, if drawn from an exponential model subject to such constraints? Will there be a similar phase transition phenomenon (as one varies the parameters) as that which occurs in the unconstrained exponential model? We present some general results for this constrained model and then apply them to get concrete answers in the edge-triangle model with fixed density of edges.
TL;DR: The proportional hazards (PH) model and its associated distributions provide suitable media for exploring connections between the Gini coefficient, Fisher information, and Shannon entropy.
Abstract: The proportional hazards (PH) model and its associated distributions provide suitable media for exploring connections between the Gini coefficient, Fisher information, and Shannon entropy. The connecting threads are Bayes risks of the mean excess of a random variable with the PH distribution and Bayes risks of the Fisher information of the equilibrium distribution of the PH model. Under various priors, these Bayes risks are generalized entropy functionals of the survival functions of the baseline and PH models and the expected asymptotic age of the renewal process with the PH renewal time distribution. Bounds for a Bayes risk of the mean excess and the Gini's coefficient are given. The Shannon entropy integral of the equilibrium distribution of the PH model is represented in derivative forms. Several examples illustrate implementation of the results and provide insights for potential applications.
TL;DR: It is proved that the martingale limits exhibit densities (bounded under suitable assumptions) and exponentially decaying tails and applications are given in the context of node degrees in random linear recursive trees and random circuits.
Abstract: In two recent works, Kuba and Mahmoud (2015a) and (2015b) introduced the family of two-color affine balanced Polya urn schemes with multiple drawings. We show that, in large-index urns (urn index between ½ and 1) and triangular urns, the martingale tail sum for the number of balls of a given color admits both a Gaussian central limit theorem as well as a law of the iterated logarithm. The laws of the iterated logarithm are new, even in the standard model when only one ball is drawn from the urn in each step (except for the classical Polya urn model). Finally, we prove that the martingale limits exhibit densities (bounded under suitable assumptions) and exponentially decaying tails. Applications are given in the context of node degrees in random linear recursive trees and random circuits.
TL;DR: This work provides a thorough study of the game version of the American call option under heterogeneous beliefs and study equilibria in randomized stopping times.
Abstract: We study zero-sum optimal stopping games (Dynkin games) between two players who disagree about the underlying model. In a Markovian setting, a verification result is established showing that if a pair of functions can be found that satisfies some natural conditions then a Nash equilibrium of stopping times is obtained, with the given functions as the corresponding value functions. In general, however, there is no uniqueness of Nash equilibria, and different equilibria give rise to different value functions. As an example, we provide a thorough study of the game version of the American call option under heterogeneous beliefs. Finally, we also study equilibria in randomized stopping times.
TL;DR: In this paper, the authors extend Goldie's implicit renewal theorem to the arithmetic case, which allows them to determine the tail behavior of the solution of various random fixed point equations, and they use the renewal theoretic approach developed by Grincevicius and Goldie.
Abstract: We extend Goldie's implicit renewal theorem to the arithmetic case, which allows us to determine the tail behavior of the solution of various random fixed point equations. It turns out that the arithmetic and nonarithmetic cases are very different. Under appropriate conditions we obtain that the tail of the solution X of the fixed point equations X =D AX + B and X =D AX ∨ B is l(x) q(x) x -κ, where q is a logarithmically periodic function q(x e h ) = q(x), x > 0, with h being the span of the arithmetic distribution of log A, and l is a slowly varying function. In particular, the tail is not necessarily regularly varying. We use the renewal theoretic approach developed by Grincevicius (1975) and Goldie (1991).
TL;DR: Two distance-based topological indices (level index and Gini index) are proposed as measures of disparity within a single tree and within tree classes and established a general expression for the level index of a tree.
Abstract: We propose two distance-based topological indices (level index and Gini index) as measures of disparity within a single tree and within tree classes. The level index and the Gini index of a single tree are measures of balance within the tree. On the other hand, the Gini index for a class of random trees can be used as a comparative measure of balance between tree classes. We establish a general expression for the level index of a tree. We compute the Gini index for two random classes of caterpillar trees and see that a random multinomial model of trees with finite height has a countable number of limits in [0, ⅓], whereas a model with independent level numbers fills the spectrum (0, ⅓].
TL;DR: The asymptotic regime in which the population size grows to ∞ and the scaled queue-length process converges to an α-stable process with a negative quadratic drift is established to characterize the head start that is needed to create a long period of uninterrupted activity (a busy period).
Abstract: We consider the Δ(i)/G/1 queue, in which a total of n customers join a single-server queue for service. Customers join the queue independently after exponential times. We consider heavy-tailed service-time distributions with tails decaying as x -α, α ∈ (1, 2). We consider the asymptotic regime in which the population size grows to ∞ and establish that the scaled queue-length process converges to an α-stable process with a negative quadratic drift. We leverage this asymptotic result to characterize the head start that is needed to create a long period of uninterrupted activity (a busy period). The heavy-tailed service times should be contrasted with the case of light-tailed service times, for which a similar scaling limit arises (Bet et al. (2015)), but then with a Brownian motion instead of an α-stable process.
TL;DR: It is demonstrated analytically for the first time that a version of TL holds for a class of distributions with infinite mean, a subset of stable laws, and the associated TL differ qualitatively from those of light-tailed distributions.
Abstract: Taylor's law (TL) originated as an empirical pattern in ecology. In many sets of samples of population density, the variance of each sample was approximately proportional to a power of the mean of that sample. In a family of nonnegative random variables, TL asserts that the population variance is proportional to a power of the population mean. TL, sometimes called fluctuation scaling, holds widely in physics, ecology, finance, demography, epidemiology, and other sciences, and characterizes many classical probability distributions and stochastic processes such as branching processes and birth-and-death processes. We demonstrate analytically for the first time that a version of TL holds for a class of distributions with infinite mean. These distributions, a subset of stable laws, and the associated TL differ qualitatively from those of light-tailed distributions. Our results employ and contribute to the methodology of Albrecher and Teugels (2006) and Albrecher et al. (2010). This work opens a new domain of investigation for generalizations of TL.
TL;DR: Upper and lower bounds are given for the asymptotic behaviour of p n,k n 1/(k+1), where pn,k is the probability of success under the optimal algorithm and n is the number of vertices exposed to a selector in some random order.
Abstract: The vertices of the kth power of a directed path with n vertices are exposed one by one to a selector in some random order. At any time the selector can see the graph induced by the vertices that have already appeared. The selector's aim is to choose online the maximal vertex (i.e. the vertex with no outgoing edges). We give upper and lower bounds for the asymptotic behaviour of p n,k n 1/(k+1), where p n,k is the probability of success under the optimal algorithm. In order to derive the upper bound, we consider a model in which the selector obtains some extra information about the edges that have already appeared. We give the exact asymptotics of the probability of success under the optimal algorithm in this case. In order to derive the lower bound, we analyse a site percolation process on a sequence of the kth powers of a directed path with n vertices.
TL;DR: In terms of monotonicity, supermodularity, and convexity of the kernel function, several sufficient conditions for the increasing convex order on the generalized aggregations are developed.
Abstract: In this paper we study general aggregation of stochastic arrangement increasing random variables, including both the generalized linear combination and the standard aggregation as special cases. In terms of monotonicity, supermodularity, and convexity of the kernel function, we develop several sufficient conditions for the increasing convex order on the generalized aggregations. Some applications in reliability and risks are also presented.
TL;DR: By treating the component availabilities over the interval as if they were availabilities at a single time point, an improved lower bound is obtained, which unlike previously given bounds, does not require the identification of all minimal path or cut vectors.
Abstract: Multistate monotone systems are used to describe technological or biological systems when the system itself and its components can perform at different operationally meaningful levels. This generalizes the binary monotone systems used in standard reliability theory. In this paper we consider the availabilities of the system in an interval, i.e. the probabilities that the system performs above the different levels throughout the whole interval. In complex systems it is often impossible to calculate these availabilities exactly, but if the component performance processes are independent, it is possible to construct lower bounds based on the component availabilities to the different levels over the interval. In this paper we show that by treating the component availabilities over the interval as if they were availabilities at a single time point, we obtain an improved lower bound. Unlike previously given bounds, the new bound does not require the identification of all minimal path or cut vectors.
TL;DR: Some limit theorems associated with the Ewens sampling formula when its parameter is increasing together with a sample size are derived.
Abstract: We derive some limit theorems associated with the Ewens sampling formula when its parameter is increasing together with a sample size.
Moreover, the limit results are applied in order to investigate asymptotic properties of the maximum likelihood estimator.
TL;DR: In this article, the authors considered the recursive Markov chain (X n ) n ≥ 0, defined by the random difference equation X n = M n X n - 1 + Q n for n ≥ 1, where X 0 is independent of (M k, Q k ) k ≥ 1.
Abstract: Given a sequence (M k , Q k ) k ≥ 1 of independent and identically distributed random vectors with nonnegative components, we consider the recursive Markov chain (X n ) n ≥ 0, defined by the random difference equation X n = M n X n - 1 + Q n for n ≥ 1, where X 0 is independent of (M k , Q k ) k ≥ 1. Criteria for the null recurrence/transience are provided in the situation where (X n ) n ≥ 0 is contractive in the sense that M 1 ⋯ M n → 0 almost surely, yet occasional large values of the Q n overcompensate the contractive behavior so that positive recurrence fails to hold. We also investigate the attractor set of (X n ) n ≥ 0 under the sole assumption that this chain is locally contractive and recurrent.
TL;DR: A large deviation principle is established for solution processes of the considered model by implementing the weak convergence technique.
Abstract: In this paper we consider a diffusive stochastic predator–prey model with a nonlinear functional response and the randomness is assumed to be of Gaussian nature. A large deviation principle is established for solution processes of the considered model by implementing the weak convergence technique.
TL;DR: The stochastic properties of the number of failed components of a three-state network made up of n components which is designed for a specific purpose according to the performance of its components is investigated.
Abstract: In this paper we investigate the stochastic properties of the number of failed components of a three-state network. We consider a network made up of n components which is designed for a specific purpose according to the performance of its components. The network starts operating at time t = 0 and it is assumed that, at any time t > 0, it can be in one of states up, partial performance, or down. We further suppose that the state of the network is inspected at two time instants t 1 and t 2 (t 1 < t 2). Using the notion of the two-dimensional signature, the probability of the number of failed components of the network is calculated, at t 1 and t 2, under several scenarios about the states of the network. Stochastic and ageing properties of the proposed failure probabilities are studied under different conditions. We present some optimal age replacement policies to show applications of the proposed criteria. Several illustrative examples are also provided.
TL;DR: By adding a vorticity matrix to the reversible transition probability matrix, it is shown that the commute time and average hitting time are smaller than that of the original reversible one.
Abstract: By adding a vorticity matrix to the reversible transition probability matrix, we show that the commute time and average hitting time are smaller than that of the original reversible one. In particular, we give an affirmative answer to a conjecture of Aldous and Fill (2002). Further quantitive properties are also studied for the nonreversible finite Markov chains.
TL;DR: This paper considers an M/G[a,b]/1/N batch service queue with bulking threshold a, max service capacity b, and buffer capacity N, and proposes a simple algorithm which guarantees to find the optimal threshold in polynomial time.
Abstract: Batch service has a wide application in manufacturing, communication networks, and cloud computing. In batch service queues with limited resources, one critical issue is to properly schedule the service so as to ensure the quality of service. In this paper we consider an M/G[a,b]/1/N batch service queue with bulking threshold a, max service capacity b, and buffer capacity N, where N can be finite or infinite. Through renewal theory, busy period analysis and decomposition techniques, we demonstrate explicitly how the bulking threshold influences the system performance such as the mean waiting time and time-averaged number of loss customers in batch service queues. We then establish a necessary and sufficient condition on the optimal bulking threshold that minimizes the expected waiting time. Enabled by this condition, we propose a simple algorithm which guarantees to find the optimal threshold in polynomial time. The performance of the algorithm is also demonstrated by numerical examples.
TL;DR: A novel law of large numbers and a central limit theorem which emerge from the nonhomogeneity of the immigration process of Sevastyanov branching processes are found.
Abstract: We consider a class of Sevastyanov branching processes with non-homogeneous Poisson immigration. These processes relax the assumption required by the Bellman-Harris process which imposes the lifespan and offspring of each individual to be independent. They find applications in studies of the dynamics of cell populations. In this paper, we focus on the subcritical case and examine asymptotic properties of the process. We establish limit theorems, which generalize classical results due to Sevastyanov and others. Our key findings include novel LLN and CLT which emerge from the non-homogeneity of the immigration process.
TL;DR: In this paper, a local martingale M associated with a Markov process was established under some restrictions on f, where Y is a process of bounded variation (on compact intervals) and either X is a jump diffusion (a special case being a Levy process) or X is some general (cadlag metric-space valued) Markov processes.
Abstract: We establish a local martingale M associate with f(X,Y) under some restrictions on f, where Y is a process of bounded variation (on compact intervals) and either X is a jump diffusion (a special case being a Levy process) or X is some general (cadlag metric-space valued) Markov process. In the latter case, f is restricted to the form f(x,y)=∑ k=1 K ξ k (x)η k (y). This local martingale unifies both Dynkin's formula for Markov processes and the Lebesgue–Stieltjes integration (change of variable) formula for (right-continuous) functions of bounded variation. For the jump diffusion case, when further relatively easily verifiable conditions are assumed, then this local martingale becomes an L 2-martingale. Convergence of the product of this Martingale with some deterministic function ( of time ) to 0 both in L 2 and almost sure is also considered and sufficient conditions for functions for which this happens are identified.