Showing papers in "Journal of Applied Probability in 2002"
TL;DR: A dual characterization of life distributions that is based on entropy applied to the past lifetime is analyzed, including its connection with the residual entropy, the relation between its increasing nature and the DRFR property, and the effect of monotonic transformations on it.
Abstract: As proposed by Ebrahimi, uncertainty in the residual lifetime distribution can be measured by means of the Shannon entropy. In this paper, we analyse a dual characterization of life distributions that is based on entropy applied to the past lifetime. Various aspects of this measure of uncertainty are considered, including its connection with the residual entropy, the relation between its increasing nature and the DRFR property, and the effect of monotonic transformations on it.
181 citations
TL;DR: In this article, the authors introduce a new concept of asymptotic independence which allows us to study the tail behavior of products and prove a limit theorem for the input process under a suitable scaling converges to a totally skewed stable Levy motion.
Abstract: The usual concept of asymptotic independence, as discussed in the context of extreme value theory, requires the distribution of the coordinatewise sample maxima under suitable centering and scaling to converge to a product measure. However, this definition is too broad to conclude anything interesting about the tail behavior of the product of two random variables that are asymptotically independent. Here we introduce a new concept of asymptotic independence which allows us to study the tail behavior of products. We carefully discuss equivalent formulations of asymptotic independence. We then use the concept in the study of a network traffic model. The usual infinite source Poisson network model assumes that sources begin data transmissions at Poisson time points and continue for random lengths of time. It is assumed that the data transmissions proceed at a constant, nonrandom rate over the entire length of the transmission. However, analysis of network data suggests that the transmission rate is also random with a regularly varying tail. So, we modify the usual model to allow transmission sources to transmit at a random rate over the length of the transmission. We assume that the rate and the time have finite mean, infinite variance and possess asymptotic independence, as defined in the paper. We finally prove a limit theorem for the input process showing that the centered cumulative process under a suitable scaling converges to a totally skewed stable Levy motion in the sense of finite-dimensional distributions.
86 citations
TL;DR: In this article, a class of continuous-time models for modeling data with heavy tails and long-range dependence is developed for stochastic volatility of asset prices and macroeconomics.
Abstract: A class of continuous-time models is developed for modelling data with heavy tails and long-range dependence. These models are based on the Green function solutions of fractional differential equations driven by Levy noise. Some exact results on the second- and higher-order characteristics of the equations are obtained. Applications to stochastic volatility of asset prices and macroeconomics are provided.
77 citations
TL;DR: In this paper, the effects of timing of payments and interest on the ruin probabilities in two generalized risk models are considered, and generalized Lundberg inequalities for the ruin probability are derived by a renewal recursive technique.
Abstract: In this paper, we study ruin probabilities in two generalized risk models. The effects of timing of payments and interest on the ruin probabilities in the models are considered. The rates of interest are assumed to have a dependent autoregressive structure. Generalized Lundberg inequalities for the ruin probabilities are derived by a renewal recursive technique. An illustrative application is given to the compound binomial risk process.
56 citations
TL;DR: In this paper, a simple way to obtain the probability generating functions for waiting time distributions of compound patterns by using the finite Markov chain imbedding method has been provided, and a computer algorithm based on Markov Chain Imbedding technique has been developed for automatically computing the distribution, probability generating function and mean of waiting time for a compound pattern.
Abstract: Probability generation functions of waiting time distributions of runs and patterns have been used successfully in various areas of statistics and applied probability. In this paper, we provide a simple way to obtain the probability generating functions for waiting time distributions of compound patterns by using the finite Markov chain imbedding method. We also study the characters of waiting time distributions for compound patterns. A computer algorithm based on Markov chain imbedding technique has been developed for automatically computing the distribution, probability generating function, and mean of waiting time for a compound pattern.
56 citations
TL;DR: In this article, the authors obtained rates of convergence to equilibrium of marked Hawkes processes in two situations: the stationary process is the empty process, in which case we speak of the rate of extinction.
Abstract: In this article we obtain rates of convergence to equilibrium of marked Hawkes processes in two situations. Firstly, the stationary process is the empty process, in which case we speak of the rate of extinction. Secondly, the stationary process is the unique stationary and nontrivial marked Hawkes process, in which case we speak of the rate of installation. The first situation models small epidemics, whereas the results in the second case are useful in deriving stopping rules for simulation algorithms of Hawkes processes with random marks.
56 citations
TL;DR: The expected Fitness of any ancestor (including the most recent common ancestor of any sample) is shown to be greater than the expected fitness of a randomly chosen gene from the population.
Abstract: We consider a nonneutral population genetics model with parent-independent mutations and two selective classes. We calculate the stationary distribution of the type of the common ancestor of a sample of genes from this model. The expected fitness of any ancestor (including the most recent common ancestor of any sample) is shown to be greater than the expected fitness of a randomly chosen gene from the population. The process of mutations to the common ancestor is also analysed. Our results are related to, but more general than, results obtained from diffusion theory.
51 citations
TL;DR: In this article, the authors study inspected systems with non-self-announcing failures where the rate of deterioration is governed by a Markov chain and compute the lifetime distribution and availability when the system is inspected according to a periodic inspection policy.
Abstract: This paper studies inspected systems with non-self-announcing failures where the rate of deterioration is governed by a Markov chain. We compute the lifetime distribution and availability when the system is inspected according to a periodic inspection policy. In doing so, we expose the role of certain transient distributions of the environment.
51 citations
TL;DR: In this paper, a simple new approach to calculate expectations of a specific form used for the pricing of derivative assets in financial mathematics is discussed, which can be found by simply integrating the respective moment generating function with a certain weight.
Abstract: We discuss a simple new approach to calculating expectations of a specific form used for the pricing of derivative assets in financial mathematics. We show that in the ‘vanilla case’, the expectations can be found by simply integrating the respective moment generating function with a certain weight. In situations corresponding to barrier-type options, we just need to carry out one more integration. The suggested approach appears to be the first (and, apart from Monte Carlo simulation, the only) one to allow the pricing of discretely monitored exotic options when the underlying asset is modelled by a general Levy process. We illustrate the method numerically by calculating the price of a discretely monitored lookback call option in the cases when the underlying follows the geometric Brownian and variance-gamma processes.
49 citations
TL;DR: In this paper, the authors give a symmetry result between the floating and fixed-strike Asian options, which involves a change of numeraire and time reversal of Brownian motion, and in this case the result allows the use of more established fixed strike pricing methods to price floating-strike options.
Abstract: There are two types of Asian options in the financial markets which differ according to the role of the average price. We give a symmetry result between the floating- and fixed-strike Asian options. The proof involves a change of numeraire and time reversal of Brownian motion. Symmetries are very useful in option valuation, and in this case the result allows the use of more established fixed-strike pricing methods to price floating-strike Asian options.
47 citations
TL;DR: In this paper, necessary and sufficient conditions are established for the process to become extinct with probability 1 and the limit probabilistic behaviour of the population size, suitably normed, is investigated.
Abstract: In this paper, the class of controlled branching processes with random control functions introduced by Yanev (1976) is considered. For this class, necessary and sufficient conditions are established for the process to become extinct with probability 1 and the limit probabilistic behaviour of the population size, suitably normed, is investigated.
TL;DR: In this paper, the authors apply coupling methods to study strong ergodicity for Markov processes, and sufficient conditions are presented in terms of the expectations of coupling times for one-dimensional diffusions and birth-death processes to be strongly ergodic.
Abstract: In this paper, we apply coupling methods to study strong ergodicity for Markov processes, and sufficient conditions are presented in terms of the expectations of coupling times. In particular, explicit criteria are obtained for one-dimensional diffusions and birth-death processes to be strongly ergodic. As a by-product, strong ergodicity implies that the essential spectra of the generators for these processes are empty.
TL;DR: In this paper, the authors introduced a bisexual Galton-Watson branching process with a mating function dependent on the population size in each generation, and investigated necessary and sufficient conditions for the process to become extinct with probability 1.
Abstract: In this paper, we introduce a bisexual Galton-Watson branching process with mating function dependent on the population size in each generation. Necessary and sufficient conditions for the process to become extinct with probability 1 are investigated for two possible conditions on the sequence of mating functions. Some results for the probability generating functions associated with the process are also given.
TL;DR: In this article, the authors studied Bernoulli bond percolation on Sierpinski carpet lattices and gave a sufficient condition for the existence of a phase transition on the lattices.
Abstract: We study Bernoulli bond percolation on Sierpinski carpet lattices, which is a class of graphs corresponding to generalized Sierpinski carpets. In this paper we give a sufficient condition for the existence of a phase transition on the lattices. The proof is suitable for graphs which have self-similarity. We also discuss the relation between the existence of a phase transition and the isoperimetric dimension.
TL;DR: In this paper, an analytical expression of the time-dependent probability distribution of M/D/1/N queues initialised in an arbitrary deterministic state is derived, and a closed form solution of the departure rate from the system is also determined.
Abstract: An analytical expression of the time-dependent probability distribution of M/D/1/N queues initialised in an arbitrary deterministic state is derived. We also obtain a simple analytical expression of the differential equation governing the transient average traffic which only involves probabilities of boundary states. As a by-product, a closed form solution of the departure rate from the system is also determined.
TL;DR: In this article, the authors considered a discrete-time queueing system with the discrete autoregressive process of order 1 (DAR(1)) as an input process and obtained the actual waiting time distribution and the virtual waiting time distributions.
Abstract: We consider a discrete-time queueing system with the discrete autoregressive process of order 1 (DAR(1)) as an input process and obtain the actual waiting time distribution and the virtual waiting time distribution. As shown in the analysis, our approach provides a natural numerical algorithm to compute the waiting time distributions, based on the theory of the GI/G/1 queue, and consequently we can easily investigate the effect of the parameters of the DAR(1) on the waiting time distributions. We also derive a simple approximation of the asymptotic decay rate of the tail probabilities for the virtual waiting time in the heavy traffic case.
TL;DR: In this paper, the authors consider the exceptional points where the above display may fail and show that the Hausdorff dimension of the exceptional sets can be obtained by introducing the right power of a logarithm.
Abstract: If Z is the branching mechanism for a supercritical Galton-Watson tree with a single progenitor and E[ Z log Z ] n at each generation n . We use the natural metric ρ(ξ,η) = e − n , where n = max{ k : ξ| k = η| k }, and observe that the local dimension index is d (μ,ξ) = lim n →∞ log(μ B (ξ| n ))/(- n ) = α = log m , for μ-almost every ξ. Our objective is to consider the exceptional points where the above display may fail. There is a nontrivial ‘thin’ spectrum for d (μ,ξ) when p 1 = P{ Z = 1} > 0 and Z has finite moments of all positive orders. Because d (μ,ξ) = a for all ξ, we obtain a ‘thick’ spectrum by introducing the ‘right’ power of a logarithm. In both cases, we find the Hausdorff dimension of the exceptional sets.
TL;DR: In this paper, a generalization of the Dirichlet process and the family of multivariate distributions it induces, with particular attention to a special case where the multivariate density function is tractable.
Abstract: The traditional Dirichlet process is characterized by its distribution on a measurable partition of the state space-namely, the Dirichlet distribution. In this paper, we consider a generalization of the Dirichlet process and the family of multivariate distributions it induces, with particular attention to a special case where the multivariate density function is tractable.
TL;DR: In this paper, the authors present a method of evaluating the distribution of a path integral for Markov chains on a countable state space, where the distribution is defined as a function of the number of chains in the state space.
Abstract: This note presents a method of evaluating the distribution of a path integral for Markov chains on a countable state space.
TL;DR: In this article, a discussion of fractional Brownian noise of index H ∈ (0, 1) is presented, and it is shown that LRD in signs holds if and only if ½ < H < 1 and LRD with respect to magnitudes in magnitudes holds if or only if ¾ ≤ H ≤ 1.
Abstract: This paper aims at enhancing the understanding of long-range dependence (LRD) by focusing on mechanisms for generating this dependence, namely persistence of signs and/or persistence of magnitudes beyond what can be expected under weak dependence. These concepts are illustrated through a discussion of fractional Brownian noise of index H ∈ (0,1) and it is shown that LRD in signs holds if and only if ½ < H < 1 and LRD in magnitudes if and only if ¾ ≤ H < 1. An application to discrimination between two risky asset finance models, the FATGBM model of Heyde and the multifractal model of Mandelbrot, is given to illustrate the use of the ideas.
TL;DR: In this paper, it was shown that the feedback flow of customers returning to the back of the queue converges to a Poisson process as the feedback delay distribution is scaled up.
Abstract: Consider a ./G/k finite-buffer queue with a stationary ergodic arrival process and delayed customer feedback, where customers after service may repeatedly return to the back of the queue after an independent general feedback delay whose distribution has a continuous density function. We use coupling methods to show that, under some mild conditions, the feedback flow of customers returning to the back of the queue converges to a Poisson process as the feedback delay distribution is scaled up. This allows for easy waiting-time approximations in the setting of Poisson arrivals, and also gives a new coupling proof of a classic highway traffic result of Breiman (1963). We also consider the case of nonindependent feedback delays.
TL;DR: In this paper, an infinite-capacity buffer receiving fluid at a rate depending on the state of an M/M/1 queue is considered and a new analytic expression for the joint stationary distribution of the buffer level and the state is obtained by the use of generating functions which are explicitly inverted.
Abstract: We consider an infinite-capacity buffer receiving fluid at a rate depending on the state of an M/M/1 queue. We obtain a new analytic expression for the joint stationary distribution of the buffer level and the state of the M/M/1 queue. This expression is obtained by the use of generating functions which are explicitly inverted. The case of a finite capacity fluid queue is also considered.
TL;DR: In this article, the stationary distribution of buffer content has a matrix exponential form, and the exponent matrix is derived using time-reversed arguments and the background state distribution at the hitting time concerning the corresponding fluid flow with upward jumps.
Abstract: We consider a fluid queue with downward jumps, where the fluid flow rate and the downward jumps are controlled by a background Markov chain with a finite state space. We show that the stationary distribution of a buffer content has a matrix exponential form, and identify the exponent matrix. We derive these results using time-reversed arguments and the background state distribution at the hitting time concerning the corresponding fluid flow with upward jumps. This distribution was recently studied for a fluid queue with upward jumps under a stability condition. We give an alternative proof for this result using the rate conservation law. This proof not only simplifies the proof, but also explains an underlying Markov structure and enables us to study more complex cases such that the fluid flow has jumps subject to a nondecreasing Levy process, a Brownian component, and countably many background states.
TL;DR: In this paper, the authors investigated k-out-of-n systems with independent and identically distributed components and obtained characterizations of the IFR(2), DMRL, NBU(2) and NBUC classes of life distributions in terms of the monotonicity of the residual life.
Abstract: In this paper, we investigate k-out-of-n systems with independent and identically distributed components. Some characterizations of the IFR(2), DMRL, NBU(2) and NBUC classes of life distributions are obtained in terms of the monotonicity of the residual life given that the (n − k)th failure has occurred at time t ≥ 0. These results complement those reported by Belzunce, Franco and Ruiz (1999). Similar conclusions based on the residual life of a parallel system conditioned by the (n − k)th failure time are presented as well.
TL;DR: In this article, the authors considered denumerable-state continuous-time Markov decision processes with (possibly unbounded) transition and reward rates and general action space under the discounted criterion and provided a set of conditions weaker than those previously known.
Abstract: In this paper, we consider denumerable-state continuous-time Markov decision processes with (possibly unbounded) transition and reward rates and general action space under the discounted criterion We provide a set of conditions weaker than those previously known and then prove the existence of optimal stationary policies within the class of all possibly randomized Markov policies Moreover, the results in this paper are illustrated by considering the birth-and-death processes with controlled immigration in which the conditions in this paper are satisfied, whereas the earlier conditions fail to hold
TL;DR: Stochastic comparisons of the number of faults obtained in partition and random testing may provide more valuable information on which testing procedures to use, and these comparisons are established for many of the well-established stochastic orders.
Abstract: Testing in order to produce software of high reliability is an area of major concern in software engineering. In an effort to find efficient methods of testing, the comparison of partition and random sampling testing methods has received considerable attention in the literature. A standard criterion for comparisons between random and partition testing, based on their expected efficacy in program debugging, is the probability of detecting at least one failure causing input in the program's domain. However, the goal in software testing is usually to find as many faults as possible in a reasonable period of time, and therefore stochastic comparisons of the number of faults obtained in partition and random testing may provide more valuable information on which testing procedures to use. We establish various conditions which guarantee that the number of faults discovered in partition testing is stochastically greater than the number discovered in random testing (using a fixed total sample size) for many of the well-established stochastic orders (including the usual stochastic order, the hazard rate order, the likelihood ratio order, and the variability order). The results established also allow us to obtain both upper and lower bounds with these stochastic orders for the sum of n independent Bernoulli random trials (with varying probability of success) in terms of the binomial distribution with parameters n and p.
TL;DR: In this article, the estimation of Markov transition matrices by Bayes' methods is considered and large and moderate deviation principles for the sequence of Bayesian posterior distributions are obtained.
Abstract: We consider the estimation of Markov transition matrices by Bayes' methods. We obtain large and moderate deviation principles for the sequence of Bayesian posterior distributions.
TL;DR: In this article, the authors consider the compound Poisson process perturbed by diffusion and derive formulae for the Laplace transform, expectation and variance of total duration of negative surplus for the CPD.
Abstract: In this paper, we consider the compound Poisson process that is perturbed by diffusion (CPD). We derive formulae for the Laplace transform, expectation and variance of total duration of negative surplus for the CPD and also present some examples of the CPD with an exponential individual claim amount distribution and a mixture exponential individual claim amount distribution.
TL;DR: In this paper, an explicit convolution representation for the equilibrium residual lifetime distribution of compound zero-modified geometric distributions is derived, and second-order reliability properties are seen to be essentially preserved under geometric compounding, and complement results of Brown (1990) and Cai and Kalashnikov (2000).
Abstract: An explicit convolution representation for the equilibrium residual lifetime distribution of compound zero-modified geometric distributions is derived Second-order reliability properties are seen to be essentially preserved under geometric compounding, and complement results of Brown (1990) and Cai and Kalashnikov (2000) The convolution representation is then extended to the nth-order equilibrium distribution This higher-order convolution representation is used to evaluate the stop-loss premium and higher stop-loss moments of the compound zero-modified geometric distribution, as well as the Laplace transform of the kth moment of the time of ruin in the classical risk model
TL;DR: In this paper, the optimal strategy which maximizes the probability of stopping on the last success is considered in a sequence of Markov-dependent trials, and the analysis is divided into two parts and both parts are realized completely.
Abstract: In a sequence of Markov-dependent trials, the optimal strategy which maximizes the probability of stopping on the last success is considered. Both homogeneous Markov chains and nonhomogeneous Markov chains are studied. For the homogeneous case, the analysis is divided into two parts and both parts are realized completely. For the nonhomogeneous case, we prove a result which contains the result of Bruss (2000) under an independence structure.