Showing papers in "Journal of Applied Probability in 1982"
TL;DR: In this article, a new Markov chain is introduced which can be used to describe the family relationships among n individuals drawn from a particular generation of a large haploid population, and the properties of this process can be studied, simultaneously for all n, by coupling techniques.
Abstract: A new Markov chain is introduced which can be used to describe the family relationships among n individuals drawn from a particular generation of a large haploid population. The properties of this process can be studied, simultaneously for all n, by coupling techniques. Recent results in neutral mutation theory are seen as consequences of the genealogy described by the chain.
1,495 citations
TL;DR: In this article, the variance and higher moments of the present value of single-stage rewards in a finite Markov decision process are presented for a semi-Markov Decision Process.
Abstract: Formulae are presented for the variance and higher moments of the present value of single-stage rewards in a finite Markov decision process. Similar formulae are exhibited for a semi-Markov decision process. There is a short discussion of the obstacles to using the variance formula in algorithms to maximize the mean minus a multiple of the standard deviation.
243 citations
TL;DR: In this article, the authors studied the relation between partial ordering and dispersion and showed that some particular pairs of distributions are ordered by dispersion, and also derived various inequalities from the results of this relation.
Abstract: Two distributions, F and G, are said be ordered in dispersion if F -1(β)-F -1(α)≦G -1(β)-G -1(α) whenever 0<α <β <1. This relation has been studied by Saunders and Moran (1978). The purpose of this paper is to study this partial ordering in detail. Few characterizations of this concept are given. These characterizations are then used to show that some particular pairs of distributions are ordered by dispersion. In addition to it some proofs of results of Saunders and Moran (1978) are simplified. Furthermore, the characterizations of this paper can be used to throw a new light on the meaning of the underlying partial ordering and also to derive various inequalities. Several examples illustrate the methods of this paper.
127 citations
TL;DR: In this paper, the threshold behavior of the stochastic spatial general epidemic model on a discrete location space is investigated by making use of the general percolation theory of McDiarmid.
Abstract: The threshold behaviour of the stochastic spatial general epidemic model on a discrete location space is investigated by making use of the general percolation theory of McDiarmid.
116 citations
TL;DR: In this article, a number of identical machines operating in parallel are used to complete the processing of a collection of jobs so as to minimize either the jobs' makespan or flowtime.
Abstract: A number of identical machines operating in parallel are to be used to complete the processing of a collection of jobs so as to minimize either the jobs' makespan or flowtime. The total processing required to complete each job has the same probability distribution, but some jobs may have received differing amounts of processing prior to the start. When the distribution has a monotone hazard rate the expected value of the makespan (flowtime) is minimized by a strategy which always processes those jobs with the least (greatest) hazard rates. When the distribution has a density whose logarithm is concave or convex these strategies minimize the makespan and flowtime in distribution. These results are also true when the processing requirements are distributed as exponential random variables with different parameters.
113 citations
TL;DR: In this paper, it was shown that if X = (X 1, · · ··, Xn ) has uniform distribution on the sphere or ball in Ω with radius a, then the joint distribution of, ···, k, converges in total variation to the standard normal distribution on ℝ.
Abstract: If X = (X 1, · ··, Xn ) has uniform distribution on the sphere or ball in ℝ with radius a, then the joint distribution of , ···, k, converges in total variation to the standard normal distribution on ℝ. Similar results hold for the inner products of independent n-vectors. Applications to geometric probability are given.
103 citations
TL;DR: In this paper, a decomposition theorem for multistate structure functions is proven and the result is applied to obtain bounds for the system performance function, which is then used to interpret the multistates structures of Barlow and Wu.
Abstract: A decomposition theorem for multistate structure functions is proven This result is applied to obtain bounds for the system performance function Another application is made to interpret the multistate structures of Barlow and Wu Various concepts of multistate importance and coherence are also discussed
98 citations
TL;DR: In this article, the authors studied the weak ergodicity of non-homogeneous Markov systems and found that the limiting structure and the relative limiting structure exist under certain conditions.
Abstract: In this paper we study the asymptotic behavior of Markov systems and especially non-homogeneous Markov systems. It is found that the limiting structure and the relative limiting structure exist under certain conditions. The problem of weak ergodicity in the above non-homogeneous systems is studied. Necessary and sufficient conditions are provided for weak ergodicity. Finally, we discuss the application of the present results in manpower systems.
90 citations
TL;DR: In this paper, an invariant probability distribution for a class of birth-and-death processes on the integers with phases and one or two boundaries was found by solving a non-linear matrix equation and then finding a probability distribution on the boundary states.
Abstract: The invariant probability distribution is found for a class of birth-and-death processes on the integers with phases and one or two boundaries. The invariant vector has a matrix geometric form and is found by solving a non-linear matrix equation and then finding an invariant probability distribution on the boundary states. Levy's concept of watching a Markov process in a subset is used to naturally decouple the computation of distributions on the boundary and interior states.
86 citations
TL;DR: In this article, a recursive formula is given, which determines the distribution of the waiting time until the sequence is obtained and leads to an expression for the probability generating function, and the mean waiting time is given until one of the given sequences is obtained, or, more generally, until a fixed number of sequences have been obtained, either different sequences or not necessarily different ones.
Abstract: Random digits are collected one at a time until a given k-digit sequence is obtained, or, more generally, until one of several k-digit sequences is obtained. In the former case, a recursive formula is given, which determines the distribution of the waiting time until the sequence is obtained and leads to an expression for the probability generating function. In the latter case, the mean waiting time is given until one of the given sequences is obtained, or, more generally, until a fixed number of sequences have been obtained, either different sequences or not necessarily different ones. Several results are known before, but the methods of proof seem to be new.
85 citations
TL;DR: It is shown that (almost) any purely inhibitory pairwise-interaction point process can be obtained in the limit of sequences of auto-logistic lattice schemes.
Abstract: Starting from a suitable sequence of auto-Poisson lattice schemes, it is shown that (almost) any purely inhibitory pairwise-interaction point process can be obtained in the limit. Further pairwise-interaction processes are obtained as limits of sequences of auto-logistic lattice schemes. SPATIAL POINT PROCESS; AUTO-POISSON SCHEME; AUTO-LOGISTIC SCHEME; STRAUSS PROCESS; PAIRWISE-INTERACTION PROCESS: INHIBITORY POINT PROCESS; HARDCORE POINT PROCESS
TL;DR: In this paper, a simple probabilistic model based on random rooted trees is proposed to assist in the identification of the number of terminal copies in a random tree, and an application to stemma construction is given.
Abstract: A fundamental task of philologists is to construct the family tree (stemma) of preserved copies of ancient manuscripts. A simple probabilistic model based on random rooted trees is proposed to assist in the identification of the number of terminal copies. The model provides the distribution of the number of terminal vertices in a random tree. An application to stemma construction is given.
TL;DR: In this article, a connection between multivariate total positivity (TP2) and multivariate monotone likelihood ratio (MLR) for probability measures on Rn is made.
Abstract: Karlin and Rinott (1980) introduced and investigated concepts of multivariate total positivity (TP2) and multivariate monotone likelihood ratio (MLR) for probability measures on Rn These TP and MLR concepts are intimately related to supermodularity as discussed in Topkis (1968), (1978) and the FKG inequality of Fortuin, Kasteleyn and Ginibre (1971). This note points out connections between these concepts and uniform conditional stochastic order (ucso) as defined in Whitt (1980). ucso holds for two probability distributions if there is ordinary stochastic order for the corresponding conditional probability distributions obtained by conditioning on subsets from a specified class. The appropriate subsets to condition on for ucso appear to be the sublattices of Rn . Then MLR implies ucso, with the two orderings being equivalent when at least one of the probability measures is TP2.
TL;DR: In this article, a class of linear birth and death processes X(t) with killing is analyzed, where the generator is of the form Ai = bi + 0, t, = ai, y, = ci, where y is the killing rate.
Abstract: We analyze a class of linear birth and death processes X(t) with killing. The generator is of the form Ai = bi + 0, t, = ai, y, = ci, where y, is the killing rate. Then P{killed in (t, t + h)JX(t) = i= yh + o(h), h 0. A variety of explicit results are found, and an example from population genetics is given.
TL;DR: In this paper, the authors considered the random uniform placement of a finite number of arcs on the circle, where the arc lengths are sampled from a distribution on (0, 1) and provided exact formulae for the probability that the circle is completely covered and for the distribution of uncovered gaps.
Abstract: Consider the random uniform placement of a finite number of arcs on the circle, where the arc lengths are sampled from a distribution on (0, 1). We provide exact formulae for the probability that the circle is completely covered and for the distribution of the number of uncovered gaps, extending Stevens's (1939) formulae for the case of fixed equal arc lengths. A special class of arc length distributions is considered, and exact probabilities of coverage are tabulated for the uniform distribution on (0, 1). Some asymptotic results for the number of gaps are also given.
TL;DR: A survey of the mathematical properties of, and the arithmetic relationships between, various distributions on the circle and the sphere can be found in this paper, where the Brownian motion and angular Gaussian distributions are shown in computer-drawn graphs to bracket the von Mises-Fisher distribution.
Abstract: A survey is made of the mathematical properties of, and the arithmetic relationships between, various distributions on the circle and the sphere. The Brownian motion and angular Gaussian distributions are shown in computer-drawn graphs to bracket the von Mises-Fisher distributions.
TL;DR: In this paper, the geometric and stochastic structure of a random Voronoi polygon relative to a homogeneous Poisson process is specified, and a key geometrical lemma elucidates the geometric structure of members of ǫ n, showing it to be simpler in one important respect than that of à − n, in that for each such N -gon of given type, there is a uniquely determined set of N generating particles.
Abstract: For each particle in an aggregate of point particles in the plane, the set of points having it as closest particle is a convex polygon, and the aggregate V of such Voronoi polygons tessellates the plane. The geometric and stochastic structure of a random Voronoi polygon relative to a homogeneous Poisson process is specified. Similarly, those points of the plane possessing the same n nearest particles constitute a convex polygon cell in the generalized Voronoi tessellation 𝒱 ( n = 2, 3, ·· ·). In fact, 𝒱 = 𝒱 1 , but to ease exposition n always takes the values 2, 3, ···. A key geometrical lemma elucidates the geometric structure of members of 𝒱 n , showing it to be simpler in one important respect than that of members of 𝒱; in that, for each such N -gon of given ‘type', there is a uniquely determined set of N generating particles. The corresponding jacobian is given, and used to derive the basic ergodic structure of 𝒱 n relative to a homogeneous Poisson process. Unlike 𝒱 no 𝒱 n contains any triangles. As n →∞, the vertices of the quadrangles of 𝒱 n tend to circularity, so that the sums of their opposite interior angles tend to π .
TL;DR: In this paper, a nonhomogeneous version of the classical record process is presented which allows two different characterizations of exponential distributions by independent non-stationary record increments, and a connection with the interarrival times of the corresponding record counting process is also pointed out.
Abstract: A non-homogeneous version of the classical record process is presented which allows two different characterizations of exponential distributions by independent non-stationary record increments. A connection with the interarrival times of the corresponding record counting process (which is pure birth) is also pointed out.
TL;DR: In this paper, the authors characterize point processes with the order statistic property without the unnecessary condition of finiteness of the first moment of the process, a condition imposed by previous researchers, and show that the class of these processes is composed only of mixed Poisson processes up to a time-scale transformation and of the mixed sample processes.
Abstract: The paper characterizes point processes with the order statistic property without the unnecessary condition of finiteness of the first moment of the process, a condition imposed by previous researchers. It shows that the class of these processes is composed only of mixed Poisson processes up to a time-scale transformation and of the mixed sample processes. It also introduces a multivariate analog of the order statistic property and characterizes completely the class of multivariate point processes with this property.
TL;DR: In this paper, the superadditive mating function for the class of Galton-Watson branching processes (GWBP) is shown to be superaddive. But the super-additive function is not sufficient for all GWBP processes.
Abstract: Mating functions considered by Asmussen (1980) and Daley (1968) for the class of bisexual Galton-Watson branching processes (GWBP) are shown to be superadditive. Consideration of a process that allows only sibling mating leads to a necessary condition for almost sure extinction in bisexual GWBP governed by superadditive mating functions. A simple example shows that the condition is not sufficient.
TL;DR: In this article, the authors studied the asymptotic behavior of self-avoiding walks with a given fraction of visits to a hyperplane when the walk is or is not constrained to be on one side of that hyperplane.
Abstract: This paper discusses the asymptotic behaviour, and in particular the critical singularities, of the generating functions for P61ya walks with a given fraction of visits to a hyperplane when the walk is or is not constrained to be on one side of that hyperplane. The three-dimensional case has applications in physics and chemistry. RANDOM WALKS; POLYMERS; SURFACE ENERGY; CRITICAL PHENOMENA Physicists and physical chemists have recently been interested in surface exponents and critical phenomena in semi-infinite systems: for example, see Bray and Moore (1977). In particular, one needs to know the number of self-avoiding walks that lie on one side of a surface and visit it a given number of times. However, self-avoiding walks are notoriously difficult to analyse; and it is natural to look at the corresponding properties for ordinary P61lya walks in the hope that these will provide some guidance for computer studies of the self-avoiding case. That hope seems to be justified in the sense that the qualitative behaviour is similar in the two cases, although the numerical constants are of course different. The present paper discusses the theory of constrained P61ya walks: some of the results are new, to the best of my knowledge, and others are refinements of previous results, such as those of Rubin (1965). Physically, interest centres on walks on the three-dimensional cubic lattice; but it is no harder to deal with the D-dimensional case, where D >2. Write X = (X,, X2, , XD) for a typical lattice point with integer coordinates. The P61ya walks are walks on this lattice with steps of unit length. We consider the following classes of walks:
TL;DR: In this paper, asymptotic properties of stereological estimators of volume fraction for stationary random sets (in the sense of Matheron) under natural and general assumptions are discussed.
Abstract: We shall discuss asymptotic properties of stereological estimators of volume (area) fraction for stationary random sets (in the sense of Matheron) under natural and general assumptions. Results obtained are strong consistency, asymptotic normality, and asymptotic unbiasedness and consistency of asymptotic variance estimators. The method is analogous to the non-parametric estimation of spectral density functions of stationary time series using window functions. Proofs are given for areal estimators, but they are also valid for lineal and point estimators with slight modifications. Finally we show that stationary Boolean models satisfy the relevant assumptions reasonably well. STEREOLOGY; VOLUME FRACTION; RANDOM SET; WINDOW FUNCTION; ASYMPTOTIC THEORY; STATIONARITY; BOOLEAN MODEL
TL;DR: In this paper, the asymptotic covariance matrix for the maximum likelihood estimates of the parameters in a cyclic Poisson model is found for the point-process case, and the close analogue with frequency estimation for Gaussian processes is emphasized.
Abstract: Suppose that it is desired to determine the frequency, known to lie in a range 0 T T increases sufficiently slowly with T, the length of the observation period, then the frequency corresponding to the maximum of the Bartlett periodogram over this range provides a consistent estimate of the unknown frequency. The asymptotic covariance matrix is found for the maximum likelihood estimates of the parameters in a cyclic Poisson model, and the close analogue with frequency estimation for Gaussian processes is emphasized. It is pointed out, however, that in the point-process case periodogram estimates will be efficient only for very special models.
Abstract: The problem initially considered is that of testing whether p = 0 in a model y(n) = x(n)+ rq(n), x(n) = px(n - 1)+ S(n) where only y(n) is observed and qr(n), S(n) are white noise. This is equivalent to distinguishing between an ARMA (1,1) model and white noise. The asymptotic distribution of the likelihood ratio criterion is derived. This is shown to be of an unusual form. This result is then used to discuss the asymptotic properties of Akaike's procedure for estimating (p, q) in an ARMA (p, q) model. If Po, q, are the true values and Po < P, qo < Q, when P, Q are the maximum values considered, then it is shown that, in a certain asymptotic sense, the procedure is sure to overestimate Po, qo. However, the asymptotic situation may be very far from that relevant in a practical case. The relevance of overestimation is briefly discussed.
TL;DR: In this article, a new measure of the importance of a component in a coherent system and derived some of its properties is proposed, for the case of components not undergoing repair proportional to the expected reduction in remaining system lifetime due to the failure of the component.
Abstract: In a previous paper, Natvig [4], we suggested a new measure of the importance of a component in a coherent system and derived some of its properties. The measure is for the case of components not undergoing repair proportional to the expected reduction in remaining system lifetime due to the failure of the component. In the present paper, we arrive at the whole distribution of this reduction in remaining system lifetime. Furthermore, for the case where components have proportional hazards, and are not repaired, a speculation of another measure is given. This measure is proportional to the derivative of the expected total lifetime of a new system with respect to the inverse of the component's proportional hazard rate.
Abstract: A survey is given of the attitude to and use of the concepts of deficient, perfect, and abundant number from the time of Nicomachus of Gerasa to that of David Anakht and Alcuin of York. Alcuin's 'Letter to Dafnin', the source of an anecdote frequently mentioned in mathematical texts, is included as an appendix. The statistics of deficient, perfect, and abundant numbers over the range 1-50000 are studied and presented graphically in several novel ways and compared with the work of Davenport and others on the distribution-law for the values of z = r(n)/(2n). Some queries are raised concerning observations which ancient writers might have been expected to make; for example, did they notice that about one half of all the even numbers are abundant?
TL;DR: The limit distribution of the k th maximum from a random sample of size n when n -+oc is identified as the distribution of lower record values from one of three extreme value distributions is given in this article.
Abstract: The limit distribution of the k th maximum from a random sample of size n when n -+,oc is identified as the distribution of the k th lower record value from one of three extreme value distributions. This fact is used in giving a different canonical representation and new proofs of the results of Hall (1978) for this limiting random variable. A characterization of the exponential distribution based on upper record values is given. MULTIVARIATE EXTREMAL DISTRIBUTION; RECORD VALUES; CHARACTERIZATION