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Showing papers in "Journal of Applied Probability in 1999"


Journal ArticleDOI
TL;DR: In this paper, the authors describe a class of asymptotic structures for the ancestral process via a convergence criterion, which prevents simultaneous mergers of ancestral lines and implies that the marginal distribution of the family size is attracted by an infinitely divisible distribution.
Abstract: Take a sample of individuals in the fixed-size population model with exchangeable family sizes Follow the ancestral lines for the sampled individuals backwards in time to observe the ancestral process We describe a class of asymptotic structures for the ancestral process via a convergence criterion One of the basic conditions of the criterion prevents simultaneous mergers of ancestral lines Another key condition implies that the marginal distribution of the family size is attracted by an infinitely divisible distribution If the latter is normal the coalescent allows only for pairwise mergers (Kingman's coalescent) Otherwise multiple mergers happen with positive probability

422 citations


Journal ArticleDOI
TL;DR: In this paper, the authors study a financial market with incompleteness arising from two sources: stochastic volatility and portfolio constraints, and derive explicit characterizations of the minimal price needed to super-replicate European-type contingent claims in this framework.
Abstract: We study a financial market with incompleteness arising from two sources: stochastic volatility and portfolio constraints. The latter are given in terms of bounds imposed on the borrowing and short-selling of a `hedger' in this market, and can be described by a closed convex set K. We find explicit characterizations of the minimal price needed to super-replicate European-type contingent claims in this framework. The results depend on whether the volatility is bounded away from zero and/or infinity, and also, on if we have linear dynamics for the stock price process, and whether volatility process depends on the stock price. We use a previously known representation of the minimal price as a supremum of the prices in the corresponding shadow markets, and we derive a PDE characterization of that representation.

137 citations


Journal ArticleDOI
TL;DR: It is shown that the paradox is avoided when resources are added across the network, rather than on a local scale, and when upgrades are focused on direct connections between the sources and destinations.
Abstract: The exponentialgrowth of computer networking demands massive upgrades in the capacity of existing networks. Traditional capacity design methodologies, developed with the single-class networking paradigm in mind, overlook the non-cooperative structure of modern networks. Consequently, such design approaches entail the danger of degraded performance when resources are added to a network, a phenomenon known as the Braess paradox. The present paper proposes methods for adding resources efficiently to a non-cooperative network of generaltopology. It is shown that the paradox is avoided when resources are added across the network, rather than on a local scale, and when upgrades are focused on direct connections between the sources and destinations. The relevance of these results for modern networks is demonstrated.

131 citations


Journal ArticleDOI
TL;DR: In this article, the authors consider limit theorems for a random walk in a random environment, (X n ), and show that the random walk satisfies a central limit theorem for any fixed environment.
Abstract: In this paper we consider limit theorems for a random walk in a random environment, (X n ). Known results (recurrence-transience criteria, law of large numbers) in the case of independent environments are naturally extended to the case where the environments are only supposed to be stationary and ergodic. Furthermore, if 'the fluctuations of the random transition probabilities around 1/2 are small', we show that there exists an invariant probability measure for 'the environments seen from the position of (X n )'. In the case of uniquely ergodic (therefore non-independent) environments, this measure exists as soon as (X n ) is transient so that the 'slow diffusion phenomenon' does not appear as it does in the independent case. Thus, under regularity conditions, we prove that, in this case, the random walk satisfies a central limit theorem for any fixed environment.

119 citations


Journal ArticleDOI
TL;DR: In this paper, a special test function was proposed to obtain sufficient conditions for the stationarity and finiteness of the moments of a general non-linear time series model, the double threshold ARMA conditional heteroskedastic (DTARMACH) model.
Abstract: Following Tweedie (1988), this paper constructs a special test function which leads to sufficient conditions for the stationarity and finiteness of the moments of a general non-linear time series model, the double threshold ARMA conditional heteroskedastic (DTARMACH) model. The results are applied to two well-known special cases, the GARCH and threshold ARMA (TARMA) models. The condition for the finiteness of the moments of the GARCH model is simple and easier to check than the condition given by Milhoj (1985) for the ARCH model. The condition for the stationarity of the TARMA model is identical to the condition given by Brockwell et al. (1992) for a special case, and verifies their conjecture that the moving average component does not affect the stationarity of the model. Under an additional irreducibility assumption, the geometric ergodicity of the GARCH and TARMA models is also established.

117 citations


Journal ArticleDOI
TL;DR: In this article, the Girsanov transformation is used to derive estimates for the accuracy of piecewise approximations for one-sided and two-sided boundary crossing probabilities using repeated numerical integration.
Abstract: Using the Girsanov transformation we derive estimates for the accuracy of piecewise approximations for one-sided and two-sided boundary crossing probabilities. We demonstrate that piecewise linear approximations can be calculated using repeated numerical integration. As an illustrative example we consider the case of one-sided and two-sided square-root boundaries for which we also present analytical representations in a form of infinite power series.

107 citations


Journal ArticleDOI
TL;DR: In this paper, the authors give new proofs of limit theorems in critical and subcritical cases of a Galton-Watson branching process conditioned on non-extinction, based on the representation of conditioned GaltonWatson generation sizes as a sum of independent increments which is derived from the decomposition of the conditioned family tree along the line of descent of the leftmost particle.
Abstract: Classical results describe the asymptotic behaviour of a Galton-Watson branching process conditioned on non-extinction. We give new proofs of limit theorems in critical and subcritical cases. The proofs are based on the representation of conditioned GaltonWatson generation sizes as a sum of independent increments which is derived from the decomposition of the conditioned Galton-Watson family tree along the line of descent of the left-most particle.

106 citations


Journal ArticleDOI
TL;DR: Asymptotically exact expressions for buffer overflow probabilities and cell loss probabilities for a finite buffer which is fed by a large number of independent and stationary sources are derived and it is shown that the results hold for a wide variety of traffic sources including ON/OFF sources with heavy-tailed distributed ON periods.
Abstract: In this paper we derive asymptotically exact expressions for buffer overflow probabilities and cell loss probabilities for a finite buffer which is fed by a large number of independent and stationary sources. The technique is based on scaling, measure change and local limit theorems and extends the recent results of Courcoubetis and Weber on buffer overflow asymptotics. We discuss the cases when the buffers are of the same order as the transmission bandwidth as well as the case of small buffers. Moreover we show that the results hold for a wide variety of traffic sources including ON/OFF sources with heavy-tailed distributed ON periods, which are typical candidates for so-called ‘self-similar’ inputs, showing that the asymptotic cell loss probability behaves in much the same manner for such sources as for the Markovian type of sources, which has important implications for statistical multiplexing. Numerical validation of the results against simulations are also reported.

102 citations


Journal ArticleDOI
TL;DR: In this paper, a subordinator model based on fractal activity time is proposed which simply explains these observed features in the data, and whose scaling properties check out well on various data sets.
Abstract: The geometric Brownian motion (Black–Scholes) model for the price of a risky asset stipulates that the log returns are i.i.d. Gaussian. However, typical log returns data shows a leptokurtic distribution (much higher peak and heavier tails than the Gaussian) as well as evidence of strong dependence. In this paper a subordinator model based on fractal activity time is proposed which simply explains these observed features in the data, and whose scaling properties check out well on various data sets.

99 citations


Journal ArticleDOI
TL;DR: In this paper, the authors consider two cases where the players are constrained to use identical strategies and use mixed strategies to minimize the expected number of steps required to meet (occupy the same node).
Abstract: Two agents are placed randomly on nodes of a known graph. They are aware of their own position, up to certain symmetries of the graph, but not that of the other agent. At each step, each agent may stay where he is or move to an adjacent node. Their common aim is to minimize the expected number of steps required to meet (occupy the same node). We consider two cases determined by whether or not the players are constrained to use identical strategies. This work extends that of Anderson and Weber on `discrete locations' (complete graph) and is related to continuous (time and space) rendezvous as formulated by Alpern. Probabilistic notions arise in the random initial placement, in the random symmetries determining spatial uncertainty of agents, and through the use of mixed strategies.

97 citations


Journal ArticleDOI
TL;DR: In this article, the distribution of the distance between words in a random sequence of letters is studied in view of application in genome sequence analysis and the exact distribution probability and cumulative distribution function of the distances between two successive occurrences of a given word and between the nth and the (n + m)th occurrences under three models of generation of the letters: i.i.d with the same probability for each letter, i.d. with different probabilities and Markov process.
Abstract: The study of the distribution of the distance between words in a random sequence of letters is interesting in view of application in genome sequence analysis. In this paper we give the exact distribution probability and cumulative distribution function of the distances between two successive occurrences of a given word and between the nth and the (n + m)th occurrences under three models of generation of the letters: i.i.d. with the same probability for each letter, i.i.d. with different probabilities and Markov process. The generating function and the first two moments are also given. The point of studying the distances instead of the counting process is that we get some knowledge not only about the frequency of a word but also about its longitudinal distribution in the sequence.

Journal ArticleDOI
TL;DR: In this paper, the authors deal with the recurrence of branching random walks on polynomially growing graphs and show that tree indexed random walks can be determined by the resistance properties of spherically symmetric graphs.
Abstract: This paper deals with the recurrence of branching random walks on polynomially growing graphs Amongst other things, we demonstrate the strong recurrence of tree indexed random walks determined by the resistance properties of spherically symmetric graphs Several branching walk models are considered to show how the branching mechanism influences the recurrence behaviour

Journal ArticleDOI
TL;DR: In this article, a large deviation principle (LDP) with an explicit rate function is proved for the estimation of drift parameter of the Ornstein-Uhlenbeck process, which is not suitable for the LDP on the score function and circumvent this key point by using a parameter-dependent change of measure.
Abstract: A large deviation principle (LDP) with an explicit rate function is proved for the estimation of drift parameter of the Ornstein-Uhlenbeck process. We establish an LDP for two estimating functions, one of them being the score function. The first one is derived by applying the Gartner–Ellis theorem. But this theorem is not suitable for the LDP on the score function and we circumvent this key point by using a parameter-dependent change of measure. We then state large deviation principles for the maximum likelihood estimator and another consistent drift estimator.

Journal ArticleDOI
TL;DR: In this article, the authors studied the asymptotic behavior of the tail of the stationary backlog distribution in a single server queue with constant service capacity c, fed by the so-called M/G/∞ input process or Cox input process.
Abstract: In this paper we study the asymptotic behavior of the tail of the stationary backlog distribution in a single server queue with constant service capacity c, fed by the so-called M/G/∞ input process or Cox input process. Asymptotic lower bounds are obtained for any distribution G and asymptotic upper bounds are derived when G is a subexponential distribution. We find the bounds to be tight in some instances, e.g. when G corresponds to either the Pareto or lognormal distribution and c - ρ < 1, where ρ is the arrival rate at the buffer.

Journal ArticleDOI
TL;DR: In this article, a strong asymptotic estimate for the queue content distribution of a fluid queue fed by a fractional Brownian input with Hurst parameter H e [1/2, 1] is studied.
Abstract: In this paper, a strong asymptotic estimate for the queue content distribution of a fluid queue fed by a fractional Brownian input with Hurst parameter H e [1/2, 1[ is studied. By applying general results on suprema of centred Gaussian processes, in particular, we show that P(V 0 > x) ≤ L/-v e -k2x2(1-H) /2 for large x. Explicit formulae for constants K, y and L are given in terms of H and system parameters.

Journal ArticleDOI
TL;DR: In this article, the authors obtained a queue length and virtual waiting time distribution for the more complicated system BMAP/SM/1 with MAP input of disasters, where the arrival of negative arrivals causes all customers to leave the system instantaneously.
Abstract: Disaster arrival in a queuing system with negative arrivals causes all customers to leave the system instantaneously. Here we obtain a queue-length and virtual waiting (sojourn) time distribution for the more complicated system BMAP/SM/1 with MAP input of disasters.

Journal ArticleDOI
TL;DR: The bifurcating autoregressive model is extended, firstly by allowing lines of descent to follow an ARMA(p, q) model rather than an AR(1) model, and secondly by allowing correlations between the environmental effects of relatives more distant than sisters.
Abstract: The bifurcating autoregressive model has been used previously to model cell lineage data. A feature of this model is that each line of descendants from an initial cell follows an AR(1) model, and that the environmental effects on sisters are correlated. However, this model concentrates on modelling the correlations between mother and daughter cells and between sister cells, and does not explain the large correlations between more distant relatives observed by some authors. Here the model is extended, firstly by allowing lines of descent to follow an ARMA(p, q) model rather than an AR(1) model, and secondly by allowing correlations between the environmental effects of relatives more distant than sisters. The models are applied to several data sets consisting of independent cell lineage trees.

Journal ArticleDOI
TL;DR: In this article, the behavior of the random triangle model on the two-dimensional triangular lattice is studied, and it is shown that phase transition occurs if and only if p = (q-1) -2/3 and q > q c for a critical value q c which turns out to equal 27 + 15√3 52.98.
Abstract: The random triangle model was recently introduced as a random graph model that captures the property of transitivity that is often found in social networks, i.e. the property that given that two vertices are second neighbors, they are more likely to be neighbors. For parameters p ∈ [0, 1] and q ≥ 1, and a finite graph G = (V, E), it assigns to elements η of {0, 1} E probabilities which are proportional to Πp e ∈ E pη(e) (1-p) 1-η(e) q t(η) , where t(η) is the number of triangles in the open subgraph. In this paper the behavior of the random triangle model on the two-dimensional triangular lattice is studied. By mapping the system onto an Ising model with external field on the hexagonal lattice, it is shown that phase transition occurs if and only if p = (q-1) -2/3 and q > q c for a critical value q c which turns out to equal 27 + 15√3 52.98. It is furthermore demonstrated that phase transition cannot occur unless p = p c (q), the critical value for percolation of open edges for given q. This implies that for q ≥ q c , p c (q) = (q - 1) -2/3 .

Journal ArticleDOI
TL;DR: The random triangle model is a Markov random graph model which, for parameters p ∊ (0, 1) and q ≥ 1 and a graph G = (V,E), assigns to a subset, η, of E, a probability which is proportional to p |η|(1-p)|E|-| η| q t(η), where t is the number of triangles in η as mentioned in this paper.
Abstract: The random triangle model is a Markov random graph model which, for parameters p ∊ (0,1) and q ≥ 1 and a graph G = (V,E), assigns to a subset, η, of E, a probability which is proportional to p |η|(1-p)|E|-|η| q t(η), where t(η) is the number of triangles in η. It is shown that this model has maximum entropy in the class of distributions with given edge and triangle probabilities. Using an analogue of the correspondence between the Fortuin-Kesteleyn random cluster model and the Potts model, the asymptotic behavior of the random triangle model on the complete graph is examined for p of order n −α, α > 0, and different values of q, where q is written in the form q = 1 + h(n) / n. It is shown that the model exhibits an explosive behavior in the sense that if h(n) ≤ c log n for c 3α, then these quantities both tend to 1. For critical values, h(n) = 3α log n + o(log n), the probability mass divides between these two extremes. Moreover, if h(n) is of higher order than log n, then the probability that η = E tends to 1, whereas if h(n) = o(log n) and α > 2/3, then, with a probability tending to 1, the resulting graph can be coupled with a graph resulting from the G(n,p) model. In particular these facts mean that for values of p in the range critical for the appearance of the giant component and the connectivity of the graph, the way in which triangles are rewarded can only have a degenerate influence.

Journal ArticleDOI
TL;DR: In this article, the shortest queue (SQ) policy minimizes various cost functions related to queue lengths and response times, and when capacities are finite the SQ policy stochastically maximizes the departure process and minimizes the loss counting process.
Abstract: We consider the problem of routeing customers to one of two parallel queues. Arrivals are independent of the state of the system but otherwise arbitrary. Assuming that queues have infinite capacities and the service times form a sequence of i.i.d. random variables with increasing likelihood ratio (ILR) distribution, we prove that the shortest queue (SQ) policy minimizes various cost functionals related to queue lengths and response times. We give a counterexample which shows that this result is not generally true when the service times have increasing hazard rate but are not increasing in the likelihood rate sense. Finally, we show that when capacities are finite the SQ policy stochastically maximizes the departure process and minimizes the loss counting process.

Journal ArticleDOI
TL;DR: In this paper, the authors introduce the notion of general final state random variables for generalized epidemic models, defined as sums over all ultimately infected individuals of random quantities of interest associated with an individual; examples include final severity.
Abstract: In this paper we introduce the notion of general final state random variables for generalized epidemic models. These random variables are defined as sums over all ultimately infected individuals of random quantities of interest associated with an individual; examples include final severity. By exploiting a construction originally due to Sellke (1983), exact results concerning the final size and general final state random variables are obtained in terms of Gontcharoff polynomials. In particular, our approach highlights the way in which these polynomials arise via simple probabilistic arguments. For ease of exposition we focus initially upon the single-population case before extending our arguments to multi-population epidemics and other variants of our basic model.

Journal ArticleDOI
TL;DR: In this paper, the distribution of generations to extinction in subcritical branching processes with Bernoulli, geometric and Poisson offspring was studied and applications to the spread of infection in highly vaccinated populations, outbreaks of enteric fever, and person-to-person transmission of human monkeypox.
Abstract: We consider the distribution of the number of generations to extinction in subcritical branching processes, with particular emphasis on applications to the spread of infectious diseases. We derive the generation distributions for processes with Bernoulli, geometric and Poisson offspring, and discuss some of their distributional and inferential properties. We present applications to the spread of infection in highly vaccinated populations, outbreaks of enteric fever, and person-to-person transmission of human monkeypox.

Journal ArticleDOI
TL;DR: In this paper, the authors give a probabilistic analogue of the mean value theorem, which is shown to be useful in various contexts of reliability theory, and provide various applications to evaluate the mean total profits of devices having random lifetimes.
Abstract: In a similar spirit to the probabilistic generalization of Taylor's theorem by Massey and Whitt [13], we give a probabilistic analogue of the mean value theorem. The latter is shown to be useful in various contexts of reliability theory. In particular, we provide various applications to the evaluation of the mean total profits of devices having random lifetimes, to the mean total-time-on-test at an arbitrary order statistic of a random sample of lifetimes, and to the mean maintenance cost for the second room of queueing systems in steady state characterized by two serial waiting rooms.

Journal ArticleDOI
TL;DR: This paper examines the connection between loss networks without controls and Markov random field theory and yields insight into the structure and computation of network equilibrium distributions, and into the nature of spatial dependence in networks.
Abstract: This paper examines the connection between loss networks without controls and Markov random field theory. The approach taken yields insight into the structure and computation of network equilibrium distributions, and into the nature of spatial dependence in networks. In addition, it provides further insight into some commonly used approximations, enables the development of more refined approximations, and permits the derivation of some asymptotically exact results.

Journal ArticleDOI
Abstract: A k-out-of-n system consisting of n components is one that works if and only if at least k of the n components work. Suppose there are n + r (1 ≤ r < n) components available of which r will be used for active redundancy. From the given n + r components, r components are chosen to be used as active redundancies, and another r components receive active redundancies (i.e. these r components are bolstered). Consider a k-out-of-n system consisting of the r bolstered and the other n - r original components. The problem of which r components should be used for active redundancy, and where to allocate them in order to maximize the lifetime of the resulting k-out-of-n system is studied. It is shown that under the usual stochastic ordering ≤st the first r weakest components should be used for active redundancy and allocated in reverse order to the next r weakest components.

Journal ArticleDOI
TL;DR: In this paper, the convergence to equilibrium of the renormalized M/M/N/N queue is analyzed and upper bounds on the distance to equilibrium are obtained and the cut-off property for two regimes of this queue is proved.
Abstract: The convergence to equilibrium of the renormalized M/M/N/N queue is analysed. Upper bounds on the distance to equilibrium are obtained and the cut-off property for two regimes of this queue is proved. Simple probabilistic methods, such as coupling techniques and martingales, are used to obtain these results.

Journal ArticleDOI
TL;DR: In this article, the authors prove uniqueness of this model in two different cases: (i) y is strictly increasing; (ii) γ(u) is differentiable for u > 0.
Abstract: For modelling non-stationary spatial random fields Z = {Z(x) : x ∈ R n ,i n ≥ 2} a recent method has been proposed to deform bijectively the index space so that the spatial dispersion D(x, y) = var[Z(x) - Z(y)], (x, y) ∈ R n x R n , depends only on the Euclidean distance in the deformed space through an isotropic variogram y. We prove uniqueness of this model in two different cases: (i) y is strictly increasing; (ii) γ(u) is differentiable for u > 0.

Journal ArticleDOI
TL;DR: In this article, the authors extended the analysis to the context of queueing systems with request repeated and showed that the limiting distribution of the system state can still be reduced to a Fredholm integral equation.
Abstract: There is a growing interest in queueing systems with negative arrivals; i.e. where the arrival of a negative customer has the effect of deleting some customer in the queue. Recently, Harrison and Pitel (1996) investigated the queue length distribution of a single server queue of type M/G/1 with negative arrivals. In this paper we extend the analysis to the context of queueing systems with request repeated. We show that the limiting distribution of the system state can still be reduced to a Fredholm integral equation. We solve such an equation numerically by introducing an auxiliary ‘truncated’ system which can easily be evaluated with the help of a regenerative approach.

Journal ArticleDOI
TL;DR: In this paper, the authors consider a discrete-time financial market model with L 1 risky asset price process subject to proportional transaction costs and derive lower and upper bounds for the limit of the super-replication cost.
Abstract: We consider a discrete-time financial market model with L 1 risky asset price process subject to proportional transaction costs. In this general setting, using a dual martingale representation we provide sufficient conditions for the super-replication cost to coincide with the replication cost. Next, we study the convergence problem in a stationary binomial model as the time step tends to zero, keeping the proportional transaction costs fixed. We derive lower and upper bounds for the limit of the super-replication cost. In the case of European call options and for a unit initial holding in the risky asset, the upper and lower bounds are equal. This result also holds for the replication cost of European call options. This is evidence (but not a proof) against the common opinion that the replication cost is infinite in a continuous-time model.

Journal ArticleDOI
TL;DR: In this article, the authors introduce open stochastic fluid networks that can be regarded as continuous analogues or fluid limits of open networks of infinite-server queues, where random exogenous input may come to any of the queues.
Abstract: We introduce open stochastic fluid networks that can be regarded as continuous analogues or fluid limits of open networks of infinite-server queues. Random exogenous input may come to any of the queues. At each queue, a c.d.f.-valued stochastic process governs the proportion of the input processed by a given time after arrival. The routeing may be deterministic (a specified sequence of successive queue visits) or proportional, i.e. a stochastic transition matrix may govern the proportion of the output routed from one queue to another. This stochastic fluid network with deterministic c.d.f.s governing processing at the queues arises as the limit of normalized networks of infinite-server queues with batch arrival processes where the batch sizes grow. In this limit, one can think of each particle having an evolution through the network, depending on its time and place of arrival, but otherwise independent of all other particles. A key property associated with this independence is the linearity: the workload associated with a superposition of inputs, each possibly having its own pattern of flow through the network, is simply the sum of the component workloads. As with infinite-server queueing models, the tractability makes the linear stochastic fluid network a natural candidate for approximations.