Showing papers in "Journal of Applied Probability in 2001"
TL;DR: In this paper, a necessary and sufficient condition in terms of σ( ∊) is given, and it is shown that when the condition fails, the behaviour of X ∊/σ(∊) can be quite intricate.
Abstract: Let X = (X(t):t ≥ 0) be a Levy process and X ∊ the compensated sum of jumps not exceeding ∊ in absolute value, σ2(∊) = var(X ∊(1)). In simulation, X - X ∊ is easily generated as the sum of a Brownian term and a compound Poisson one, and we investigate here when X ∊/σ(∊) can be approximated by another Brownian term. A necessary and sufficient condition in terms of σ(∊) is given, and it is shown that when the condition fails, the behaviour of X ∊/σ(∊) can be quite intricate. This condition is also related to the decay of terms in series expansions. We further discuss error rates in terms of Berry-Esseen bounds and Edgeworth approximations.
265 citations
IBM1
TL;DR: In this article, an optimal stopping time problem for Russian options under the hidden Markov process was investigated. But the optimal strategy involves jumping over the optimal boundary and by an arbitrary overshoot, which is a nonstandard application of the well-known smooth fit principle.
Abstract: We investigate an optimal stopping time problem which arises from pricing Russian options (i.e. perpetual look-back options) on a stock whose price fluctuations are modelled by adjoining a hidden Markov process to the classical Black-Scholes geometric Brownian motion model. By extending the technique of smooth fit to allow jump discontinuities, we obtain an explicit closed-form solution. It gives a non-standard application of the well-known smooth fit principle where the optimal strategy involves jumping over the optimal boundary and by an arbitrary overshoot. Based on the optimal stopping analysis, an arbitrage-free price for Russian options under the hidden Markov model is derived.
180 citations
TL;DR: In this paper, the authors study random variables related to a shock reliability model and obtain properties of the distribution function of the random variables involved and obtain their limit behavior when k tends to infinity or when the probability of entering a critical set tends to zero.
Abstract: In this paper we study random variables related to a shock reliability model. Our models can be used to study systems that fail when k consecutive shocks with critical magnitude (e.g. above or below a certain critical level) occur. We obtain properties of the distribution function of the random variables involved and we obtain their limit behaviour when k tends to infinity or when the probability of entering a critical set tends to zero. This model generalises the Poisson shock model.
149 citations
TL;DR: In this treatment, a characterization of the signature of a system with independent identically distributed components is given in terms of the number of path sets in the system as well as in levels of what are called ordered cut sets.
Abstract: If r is the lifetime of a coherent system, then the signature of the system is the vector of probabilities that the lifetime coincides with the ith order statistic of the component lifetimes. The signature can be useful in comparing different systems. In this treatment we give a characterization of the signature of a system with independent identically distributed components in terms of the number of path sets in the system as well as in terms of the number of what we call ordered cut sets. We consider, in particular, the signatures of indirect majority systems and compare them with the signatures of simple majority systems of the same size. We note that the signature of an indirect majority system of size r x s = n is symmetric around 1(n + 1), and use this to show that the expected lifetime of an r x s = n indirect majority system exceeds that of a simple (direct) majority system of size n when the components are exponentially distributed with the same parameter.
137 citations
TL;DR: In this article, the authors used the modified Gutenberg-Richter law to model the sizes of earthquakes and found that the maximum likelihood estimates of the cutoff parameter are substantially biased. And they presented alternative estimates for the cutoff parameters and their properties discussed.
Abstract: The tapered (or generalized) Pareto distribution, also called the modified Gutenberg--Richter law, has been used to model the sizes of earthquakes. Unfortunately, maximum likelihood estimates of the cutoff parameter are substantially biased. Alternative estimates for the cutoff parameter are presented, and their properties discussed.
113 citations
TL;DR: In this article, Wang and Potzelberger derived an explicit formula for the probability that a Brownian motion crosses a one-sided piecewise linear boundary and used this formula to approximate the boundary crossing probability for general nonlinear boundaries.
Abstract: Wang and Potzelberger (1997) derived an explicit formula for the probability that a Brownian motion crosses a one-sided piecewise linear boundary and used this formula to approximate the boundary crossing probability for general nonlinear boundaries. The present paper gives a sharper asymptotic upper bound of the approximation error for the formula, and generalizes the results to two-sided boundaries. Numerical computations are easily carried out using the Monte Carlo simulation method. A rule is proposed for choosing optimal nodes for the approximating piecewise linear boundaries, so that the corresponding approximation errors of boundary crossing probabilities converge to zero at a rate of O(1/n 2 ).
109 citations
TL;DR: In this paper, the impact of a broadly understood trend, which includes a change point in mean and monotonic trends studied by Bhattacharya et al. (1983), on the asymptotic behaviour of a class of tests designed to detect long memory in a stationary sequence was studied.
Abstract: The paper studies the impact of a broadly understood trend, which includes a change point in mean and monotonic trends studied by Bhattacharya et al. (1983), on the asymptotic behaviour of a class of tests designed to detect long memory in a stationary sequence. Our results pertain to a family of tests which are similar to Lo's (1991) modified R/S test. We show that both long memory and nonstationarity (presence of trend or change points) can lead to rejection of the null hypothesis of short memory, so that further testing is needed to discriminate between long memory and some forms of nonstationarity. We provide quantitative description of trends which do or do not fool the R/S-type long memory tests. We show, in particular, that a shift in mean of a magnitude larger than N -1/2 , where N is the sample size, affects the asymptotic size of the tests, whereas smaller shifts do not do so.
106 citations
TL;DR: In this paper, the genealogical structure of samples from a population for which any given generation is made up of direct descendants from several previous generations was studied and weak convergence of the scaled ancestral process, with the usual diffusion scaling, to a coalescent process which is equivalent to a time-changed version of Kingman's coalescent was shown.
Abstract: We study the genealogical structure of samples from a population for which any given generation is made up of direct descendants from several previous generations. These occur in nature when there are seed banks or egg banks allowing an individual to leave offspring several generations in the future. We show how this temporal structure in the reproduction mechanism causes a decrease in the coalescence rate. We also investigate the effects of age-dependent neutral mutations. Our main result gives weak convergence of the scaled ancestral process, with the usual diffusion scaling, to a coalescent process which is equivalent to a time-changed version of Kingman's coalescent.
99 citations
TL;DR: In this paper, the authors prove the statement of the title and show that it has useful applications in evaluating the convergence of queueing models and Gibbs samplers with deterministic and random scans.
Abstract: The paper proves the statement of the title, and shows that it has useful applications in evaluating the convergence of queueing models and Gibbs samplers with deterministic and random scans.
91 citations
IBM1
TL;DR: In this article, the authors studied the optimal stopping problem for different kinds of options, based on the Black-Scholes model of stock fluctuations, and showed that the optimal policy depends on the ratio of x/l, where x is the current stock price.
Abstract: We solve the following three optimal stopping problems for different kinds of options, based on the Black–Scholes model of stock fluctuations. (i) The perpetual lookback American option for the running maximum of the stock price during the life of the option. This problem is more difficult than the closely related one for the Russian option, and we show that for a class of utility functions the free boundary is governed by a nonlinear ordinary differential equation. (ii) A new type of stock option, for a company, where the company provides a guaranteed minimum as an added incentive in case the market appreciation of the stock is low, thereby making the option more attractive to the employee. We show that the value of this option is given by solving a nonalgebraic equation. (iii) A new call option for the option buyer who is risk-averse and gets to choose, a priori, a fixed constant l as a ‘hedge’ on a possible downturn of the stock price, where the buyer gets the maximum of l and the price at any exercise time. We show that the optimal policy depends on the ratio of x/l , where x is the current stock price.
81 citations
TL;DR: In this paper, the existence of critical Hawkes point processes with a finite average intensity was proved under a heavy-tail condition for the fertility rate which is related to a long-range dependence property.
Abstract: In this article, we prove the existence of critical Hawkes point processes with a finite average intensity, under a heavy-tail condition for the fertility rate which is related to a long-range dependence property. Criticality means that the fertility rate integrates to 1, and corresponds to the usual critical branching process, and, in the context of Hawkes point processes with a finite average intensity, it is equivalent to the absence of ancestors. We also prove an ergodic decomposition result for stationary critical Hawkes point processes as a mixture of critical Hawkes point processes, and we give conditions for weak convergence to stationarity of critical Hawkes point processes.
TL;DR: In this article, a hidden semi-Markov model for breakpoint rainfall data is proposed, which consists of both the times at which rain-rate changes and the steady rates between such changes.
Abstract: The paper proposes a hidden semi-Markov model for breakpoint rainfall data that consist of both the times at which rain-rate changes and the steady rates between such changes. The model builds on and extends the seminal work of Ferguson (1980) on variable duration models for speech. For the rainfall data the observations are modelled as mixtures of log-normal distributions within unobserved states where the states evolve in time according to a semi-Markov process. For the latter, parametric forms need to be specified for the state transition probabilities and dwell-time distributions. Recursions for constructing the likelihood are developed and the EM algorithm used to fit the parameters of the model. The choice of dwell-time distribution is discussed with a mixture of distributions over disjoint domains providing a flexible alternative. The methods are also extended to deal with censored data. An application of the model to a large-scale bivariate dataset of breakpoint rainfall measurements at Wellington, New Zealand, is discussed.
TL;DR: Two burn-in procedures for a general failure model where the failed component is repaired completely regardless of the type of failure are considered and the problems of determining optimal burn- in time and optimal replacement policy are considered.
Abstract: In this paper two burn-in procedures for a general failure model are considered. There are two types of failure in the general failure model. One is Type I failure (minor failure) which can be removed by a minimal repair or a complete repair and the other is Type II failure (catastrophic failure) which can be removed only by a complete repair. During a burn-in process, with burn-in Procedure I, the failed component is repaired completely regardless of the type of failure, whereas, with burn-in Procedure II, only minimal repair is done for the Type I failure and a complete repair is performed for the Type II failure. In field use, the component is replaced by a new burned-in component at the ‘field use age’ T or at the time of the first Type II failure, whichever occurs first. Under the model, the problems of determining optimal burn-in time and optimal replacement policy are considered. The two burn-in procedures are compared in cases when both the procedures are applicable.
TL;DR: The aim of the paper is to show how point process techniques can be used to derive the asymptotic behavior of the sample autocorrelation function of this process with heavy-tailed marginal distributions.
Abstract: The paper considers one of the standard processes for modeling returns in finance, the stochastic volatility process with regularly varying innovations. The aim of the paper is to show how point process techniques can be used to derive the asymptotic behavior of the sample autocorrelation function of this process with heavy-tailed marginal distributions. Unlike other non-linear models used in finance, such as GARCH and bilinear models, sample autocorrelations of a stochastic volatility process have attractive asymptotic properties. Specifically, in the infinite variance case, the sample autocorrelation function converges to zero in probability at a rate that is faster the heavier the tails of the marginal distribution. This behavior is analogous to the asymptotic behavior of the sample autocorrelations of independent identically distributed random variables.
TL;DR: In this article, the mean time to extinction for the stochastic logistic model has been studied in various contexts, including epidemiology, population biology, chemistry and sociology, and a number of approximation formulae for these quantities have been derived.
Abstract: The stochastic logistic model has been studied in various contexts, including epidemiology, population biology, chemistry and sociology. Among the model predictions, the quasistationary distribution and the mean time to extinction are of major interest for most applications, and a number of approximation formulae for these quantities have been derived. In this paper, previous approximation formulae are improved for two mathematically tractable cases: at the limit of the number of individuals N → ∞ (with relative error of the approximations of the order O(1/N)), and at the limit of the basic reproduction ratio R 0 → ∞ (with relative error of the approximations of the order O(1/R 0 )). The mathematically rigorous formulae are then extended heuristically for other values of N and R 0 > 1.
TL;DR: In this paper, the authors consider Markov chains in the context of iterated random functions and show the existence and uniqueness of an invariant distribution under a local contraction condition combined with a drift condition, extending results of Diaconis and Freedman.
Abstract: We consider Markov chains in the context of iterated random functions and show the existence and uniqueness of an invariant distribution under a local contraction condition combined with a drift condition, extending results of Diaconis and Freedman. From these we deduce various other topological stability properties of the chains. Our conditions are typically satisfied by, for example, queueing and storage models where the global Lipschitz condition used by Diaconis and Freedman normally fails.
TL;DR: In this article, the authors present a new inspection policy useful when testing is needed to detect failures of a single-unit system and study the behavior of the optimum policy for some time to failure distributions often assumed in reliability: exponential and Pareto.
Abstract: We present a new inspection policy useful when testing is needed to detect failures of a single-unit system. It is supposed that tests may fail and give an erroneous result. The inspection policy minimizing cost per unit of time for an infinite time span is also discussed. In addition, we study the behaviour of the optimum policy for some time to failure distributions often assumed in reliability: exponential and Pareto.
TL;DR: In this paper, a finite Markov chain imbedding method is developed for the study of X r,e in the case of the non-overlapping and overlapping way of counting runs and patterns.
Abstract: Let {X n , n ≥ 1} be a sequence of trials taking values in a given set A, let e be a pattern (simple or compound), and let X r,e be a random variable denoting the waiting time for the rth occurrence of e. In the present article a finite Markov chain imbedding method is developed for the study of X r,e in the case of the non-overlapping and overlapping way of counting runs and patterns. Several extensions and generalizations are also discussed.
TL;DR: In this article, an explicit and computable criterion for strong ergodicity of single-birth processes is presented, and sufficient conditions are obtained for strong Ergodicity for an extended class of continuous-time branching processes and multi-dimensional Q-processes by comparison methods.
Abstract: An explicit and computable criterion for strong ergodicity of single-birth processes is presented. As an application, some sufficient conditions are obtained for strong ergodicity of an extended class of continuous-time branching processes and multi-dimensional Q-processes by comparison methods respectively. Consequently strong ergodicity of the Q-process corresponding to the finite-dimensional Schlogl model is proven.
TL;DR: In this paper, a simple and effective formula for the distribution of S n (r) via the finite Markov chain embedding technique was derived to overcome some of the limitations of the existing complex formulae.
Abstract: Let be the scan statistic of window size r for a sequence of n bistate trials . The scan statistic S n (r) has been successfully used in various fields of applied probability and statistics, and its distribution has been studied extensively in the literature. Currently, all existing formulae for the distribution of S n (r) are rather complex, and they can only be numerically implemented when is a sequence of Bernoulli trials, the window size r is less than 20 and the length of the sequence n is not too large. Hence, these formulae have been limiting the practical applications of the scan statistic. In this article, we derive a simple and effective formula for the distribution of S n (r) via the finite Markov chain embedding technique to overcome some of the limitations of the existing complex formulae. This new formula can be applied when is either a sequence of Bernoulli trials or a sequence of Markov dependent bistate trials. Selected numerical examples are given to illustrate our results.
TL;DR: For non-compact state spaces, the method is still sound, but convergence results are scarce as discussed by the authors, and it is difficult to prove convergence in such cases, for Markov chains satisfying suitable drift and minorization conditions.
Abstract: Simulated annealing is a popular and much studied method for maximizing functions on finite or compact spaces. For noncompact state spaces, the method is still sound, but convergence results are scarce. We show here how to prove convergence in such cases, for Markov chains satisfying suitable drift and minorization conditions.
TL;DR: In this paper, the instantaneous availability of a system maintained under periodic inspection is investigated using random walk models, and the relationship between this problem and the random walk model in a two-dimensional plane is described.
Abstract: In this paper, the instantaneous availability of a system maintained under periodic inspection is investigated using random walk models. Two cases are considered. In the first model, the system is repaired or modified and it is assumed to be as good as new upon periodic inspection and maintenance. In the second model, the system is not modified after the inspection if the system is still working, and the condition of the system is assumed to be the same as that before the inspection. For both models the failures only can be found through the inspection. Perfect repair or replacement of a failed system is assumed to be carried out, but the time it takes can be constant or of a random length. The relationship between this problem and the random walk model in a two-dimensional plane is described. Several new results are also shown.
TL;DR: In this paper, the authors review the formulation of the linked stress release model for large scale seismicity together with aspects of its application, using data from Taiwan for illustrative purposes, using tools that include Akaike's information criterion (AIC), numerical analysis, residual point processes and Monte Carlo simulation.
Abstract: The paper reviews the formulation of the linked stress release model for large scale seismicity together with aspects of its application. Using data from Taiwan for illustrative purposes, models can be selected and verified using tools that include Akaike's information criterion (AIC), numerical analysis, residual point processes and Monte Carlo simulation.
TL;DR: In this paper, the Laplace-Stieltjes transform of the busy period of a multi-server M/M/c queueing system is used to compute the moments of the length of a busy period and the number of customers served during it.
Abstract: This paper presents an algorithmic analysis of the busy period for the M/M/c queueing system. By setting the busy period equal to the time interval during which at least one server is busy, we develop a first step analysis which gives the Laplace-Stieltjes transform of the busy period as the solution of a finite system of linear equations. This approach is useful in obtaining a suitable recursive procedure for computing the moments of the length of a busy period and the number of customers served during it. The maximum entropy formalism is then used to analyse what is the influence of a given set of moments on the distribution of the busy period and to estimate the true busy period distribution. Our study supplements a recent work of Daley and Servi (1998) and other studies where the busy period of a multiserver queue follows a different definition, i.e., a busy period is the time interval during which all servers are engaged.
Abstract: The coupon subset collection problem is a generalization of the classical coupon collecting problem, in that rather than collecting individual coupons we obtain, at each time point, a random subset of coupons. The problem of interest is to determine the expected number of subsets needed until each coupon is contained in at least one of these subsets. We provide bounds on this number, give efficient simulation procedures for estimating it, and then apply our results to a reliability problem.
TL;DR: In this paper, it was shown that the NBU(2) class of life distributions is closed under the formation of series systems and under convolution operation, and also under series systems, it was proved that the NB2 class is closed in the case of the increasing concave ordering of the excess lifetime.
Abstract: Some new results about the NBU(2) class of life distributions are obtained. Firstly, it is proved that the decreasing with time of the increasing concave ordering of the excess lifetime in a renewal process leads to the NBU(2) property of the interarrival times. Secondly, the NBU(2) class of life distributions is proved to be closed under the formation of series systems. Finally, it is also shown that the NBU(2) class is closed under convolution operation.
TL;DR: In this article, the authors studied the travel time needed to pick a list of items when the carousel operates under the nearest item heuristic and found a closed form expression for the distribution and all moments of the travel times.
Abstract: A carousel is an automated warehousing system consisting of a large number of drawers rotating in a closed loop. In this paper, we study the travel time needed to pick a list of items when the carousel operates under the nearest item heuristic. We find a closed form expression for the distribution and all moments of the travel time. We also analyse the asymptotic behaviour of the travel time when the number of items tends to infinity. All results follow from probabilistic arguments based on properties of uniform order statistics.
TL;DR: The average R score of the annual predictions in China in the period 1990-1998 is about 0.184, significantly larger than 0.0 as mentioned in this paper, and the nine-year mean R score is only marginally higher than this background value.
Abstract: The annual earthquake predictions of the China Seismological Bureau (CSB) are evaluated by means of an R score (an R score is approximately 0 for completely random guesses, and approximately 1 for completely successful predictions). The average R score of the annual predictions in China in the period 1990–1998 is about 0.184, significantly larger than 0.0. However, background seismicity is higher in seismically active regions. If a ‘random guess' prediction is chosen to be proportional to the background seismicity, the expected R score is 0.123, and the nine-year mean R score of 0.184 as observed is only marginally higher than this background value. Monte Carlo tests indicate that the probability of attaining an R score of actual prediction by background seismicity based on random guess is about . It is concluded that earthquake prediction in China is still in a very preliminary stage, barely above a pure chance level.
TL;DR: In this article, the optimal stopping value of random variables X l,..., X n depends on the joint distribution function of the random variables and hence on their marginals as well as on their dependence structure.
Abstract: The optimal stopping value of random variables X l,..., X n depends on the joint distribution function of the random variables and hence on their marginals as well as on their dependence structure. The maximal and minimal values of the optimal stopping problem is determined within the class of all joint distributions with fixed marginals F l,..., F n . They correspond to some sort of strong negative or positive dependence of the random variables. Any value inbetween these two extremes is attained for some dependence structures. The determination of the minimal value is based on some new ordering results for probability measures, in particular on lattice properties of stochastic orderings. We also identify properties of dependence structures leading to the minimal optimal stopping value. In the proofs we need an extension of Strassen's theorem on representation of the convex order which reveals that convex ordered distributions can be coupled by a two-step martingale (X, Y) with the additional property that Y is stochastically increasing in X.
TL;DR: In this paper, it was shown that κ (min( X, Y )) ≥ κ( X ) + λ( Y ) for independent non-negative RVs X and Y.
Abstract: For a random variable (RV) X its moment index κ ( X ) ≡ sup{ κ : E(| X | κ ) κ ( X ) ≤ ∞; it is a critical quantity and finite for heavy-tailed RVs The paper shows that κ (min( X, Y )) ≥ κ ( X ) + κ ( Y ) for independent non-negative RVs X and Y For independent non-negative ‘excess' RVs X s and Y s whose distributions are the integrated tails of X and Y, κ ( X ) + κ ( Y ) ≤ κ (min( X s , Y s )) + 2 ≤ κ (min( X, Y )) An example shows that the inequalities can be strict, though not if the tail of the distribution of either X or Y is a regularly varying function