Showing papers in "Journal of Applied Probability in 1974"
TL;DR: In this paper, it was shown that all stationary self-exciting point processes with finite intensity may be represented as Poisson cluster processes which are age-dependent immigration-birth processes, and their existence was established.
Abstract: It is shown that all stationary self-exciting point processes with finite intensity may be represented as Poisson cluster processes which are age-dependent immigration-birth processes, and their existence is established. This result is used to derive some counting and interval properties of these processes using the probability generating functional.
709 citations
TL;DR: In this paper, the authors generalize the results obtained for models which assume a discrete state-space or discrete time or both, to a model with both state space and time continuous.
Abstract: Results on the behaviour of Markov branching processes as time goes to infinity, hitherto obtained for models which assume a discrete state-space or discrete time or both, are here generalised to a model with both state-space and time continuous. The results are similar but the methods not always so.
180 citations
TL;DR: In this article, the exponential limiting formula for the waiting time distribution function in heavy traffic, conjectured by Kingman (1965) and established by Kollerstrom (1974), is extended and further approximations for the characteristic function and error bounds for the limiting foemulae are investigated.
Abstract: The queues being studied here are of the type GI/G/k in statistical equilibrium (with traffic intensity less than one). The exponential limiting formula for the waiting time distribution function in heavy traffic, conjectured by Kingman (1965) and established by Kollerstrom (1974), is extended here. The asymptotic properties of the moments are investigated as well as further approximations for the characteristic function and error bounds for the limiting foemulae.
101 citations
TL;DR: In this paper, the determinant and inverse of the covariance matrix of a set of n consecutive observations on a mixed autoregressive moving average process are given for the general autoregression process of order p (n? p), and for the first order mixed auto-regression process.
Abstract: Expressions are obtained for the determinant and inverse of the covariance matrix of a set of n consecutive observations on a mixed autoregressive moving average process. Explicit formulae for the inverse of this matrix are given for the general autoregressive process of order p (n ? p), and for the first order mixed autoregressive moving average process.
96 citations
TL;DR: This paper shows that an unrooted tree with labelled terminal vertices may provide a better representation of the relationships between the objects because the similarity indices are required to conform to fewer restrictions.
Abstract: It is common to represent taxonomic hierarchies of related objects (such as similar plant or animal species or languages of the same family) by rooted trees with labelled terminal vertices which represent the objects. The multivariate data comparing numerous characteristics of the objects is first reduced to indices of similarity (or more often of dissimilarity) between each pair of objects. These are used to classify the objects into groups which are then depicted on a tree. This paper shows that an unrooted tree with labelled terminal vertices may provide a better representation of the relationships between the objects because the similarity indices are required to conform to fewer restrictions. Also for a given number of terminal vertices, there are fewer unrooted than rooted trees so that studies using probability distributions of trees or seeking the most suitable tree to represent the data are more practicable.
89 citations
TL;DR: In this article, the characteristic function (ch.f.) ϕ 0 of Z 1 is Lebesgue-integrable over (∞, ∞) and the identity is to be understood in the sense of convergence in distribution.
Abstract: Let {Zt ; t = 0, ± 1, ···} be a pure white noise process with γ = E{|Z 1 | δ} 0. Assume that the characteristic function (ch.f.) ϕ 0 of Z 1 is Lebesgue-integrable over (—∞, ∞). Let {gv ;v = 0, 1, 2, ···, g 0 = 1} be a sequence of real numbers such that where λ = δ(1 + δ)−1. Define , where the identity is to be understood in the sense of convergence in distribution. Then {Xt ; t = 0, ± 1, ···} is a strongly mixing stationary process in the sense that if is the σ-field generated by the random variables (r.v.) Xa , ···, Xb then for any where M is a finite positive constant which depends only on ϕ 0 and
84 citations
TL;DR: The accurate space derivative (ASD) method for the numerical treatment of nonlinear partial differential equations has been applied to the solution of Fisher's equation, a nonlinear diffusion equation describing the rate of advance of a new advantageous gene, and which is also related to certain water waves and plasma shock waves described by the Korteweg-de-Vries-Burgers equation as discussed by the authors.
Abstract: The accurate space derivative (ASD) method for the numerical treatment of nonlinear partial differential equations has been applied to the solution of Fisher's equation, a nonlinear diffusion equation describing the rate of advance of a new advantageous gene, and which is also related to certain water waves and plasma shock waves described by the Korteweg-de-Vries-Burgers equation. The numerical experiments performed indicate how from a variety of initial conditions, (including a step function, and a wave with local perturbation) the concentration of advantageous gene evolves into the travelling wave of minimal speed. For an initial superspeed wave this evolution depends on the cutting off of the right-hand tail of the wave, which is physically plausible; this condition is necessary for the convergence of the ASD method. Detailed comparisons with an analytic solution for the travelling waves illustrate the striking accuracy of the ASD method for other than very small values of the concentration. NONLINEAR DIFFUSION; EPIDEMIC WAVES; NUMERICAL SIMULATION
84 citations
TL;DR: In this article, bounds for the limiting distribution of the delay in queue for a GI/G/1 system via Martingale theory were obtained for the special cases where the service distribution is either IFR, DFR, NBU or NWU.
Abstract: : Bounds are obtained for the limiting distribution of the delay in queue for a GI/G/1 system via Martingale theory. These bounds are somewhat stronger than similar bounds recently obtained by Kingman. Simplifications of the bounds are obtained in the special cases where the service distribution is either IFR, DFR, NBU or NWU.
83 citations
TL;DR: Galton-Watson processes where reproduction of individuals in different generations can have different distributions retain many characteristic features of classical processes as mentioned in this paper, and they have been shown to retain many of the characteristics of classical Galton Watson processes.
Abstract: Galton-Watson processes where the reproduction of individuals in different generations can have different distributions retain many characteristic features of classical processes.
82 citations
TL;DR: In this paper, the concomitants of the order statistics, which arise naturally in selection procedures based on the Xr :n, were shown to be independent, identically distributed variates.
Abstract: In a random sample of n pairs (X r , Y r ), r = 1, 2, …, n, drawn from a bivariate normal distribution, let Xr :n be the rth order statistic among the Xr and let Y [r:n] be the Y-variate paired with Xr :n . The Y[r:n] , which we call concomitants of the order statistics, arise most naturally in selection procedures based on the Xr :n . It is shown that asymptotically the k quantities k fixed, are independent, identically distributed variates. In addition, putting Rt,n for the number of integers j for which , the asymptotic distribution and all moments of n– 1 Rt, n are determined for t such that t/n → λ with 0 < λ < 1.
77 citations
TL;DR: In this paper, the number of mutants is shown to be a stable-Levy random variable, whose exponent α is essentially the ratio of the growth rates of non-mutants and of mutants.
Abstract: Luria and Delbruck (1943) have observed that, in old cultures of bacteria that have mutated at random, the distribution of the number of mutants is extremely long-tailed. In this note, this distribution will be derived (for the first time) exactly and explicitly. The rates of mutation will be allowed to be either positive or infinitesimal, and the rate of growth for mutants will be allowed to be either equal, greater or smaller than for non-mutants. Under the realistic limit condition of a very low mutation rate, the number of mutants is shown to be a stable-Levy (sometimes called “Pareto Levy”) random variable, of maximum skewness s, whose exponent α is essentially the ratio of the growth rates of non-mutants and of mutants. Thus, the probability of the number of mutants exceeding the very large value m is proportional to m –α–1 (a behavior sometimes referred to as “asymptotically Paretian” or “hyperbolic”). The unequal growth rate cases α ≠ 1 are solved for the first time. In the α = 1 case, a result of Lea and Coulson is rederived, interpreted, and generalized. Various paradoxes involving divergent moments that were encountered in earlier approaches are either absent or fully explainable. The mathematical techniques used being standard, they will not be described in detail, so this note will be primarily a collection of results. However, the justification for deriving them lies in their use in biology, and the mathematically unexperienced biologists may be unfamiliar with the tools used. They may wish for more details of calculations, more explanations and Figures. To satisfy their needs, a report available from the author upon request has been prepared. It will be referred to as Part II.
TL;DR: In this paper, a sequence of independent, identically distributed random variables with EXI = p n converges weakly to a limit random variable, S*, and to find the Laplace transform of the distribution of S*,.
Abstract: Let {Xk: k ? 1 } be a sequence of independent, identically distributed random variables with EXI = p n converges weakly to a limit random variable, S*, and to find the Laplace transform of the distribution of S*. We also investigate a collection of random walks with mean p < 0 and conditional limits S* (p), and show that S* (p), properly normalized, converges to a gamma distribution of second order as s /; 0. These results have applications to the GI/G/1 queue, collective risk theory, and the gambler's ruin problem.
TL;DR: In this article, the problem of determining optimal release rates for finite dams when the release rate may take one of two values and specific costs are involved in increasing and decreasing this release rate is discussed.
Abstract: The problem discussed is that of determining optimal release rates for finite dams when the release rate may take one of two values and specific costs are involved in increasing and decreasing this release rate. From another point of view, this is a problem of controlling the output simply by switching on and off. A diffusion model is suggested and by considering a family of plausible output policies an optima procedure, which minimises the long-term average cost of operating the dam, is established. Various properties of this output policy are given.
TL;DR: In this article, the authors introduce the "delay" time for equipment replacement, derive the expected cost per unit time, and discuss the optimum replacement policies under several conditions under several special cases.
Abstract: Replacement theory for equipment has been investigated by several authors. This paper introduces the 'delay' time preparing for replacement, derives the expected cost per unit time, and discusses the optimum replacement policies under several conditions. Three special cases are discussed and numerical examples are presented. REPLACEMENT PROBLEMS WITH DELAY; RENEWAL PROCESSES; COSTS; SCHEDULED REPLACEMENT; OPTIMIZATION
TL;DR: In this article, the classical secretary problem is generalized to admit stochastically successful procurement of previous interviewees, but each has a certain probability of refusing the offer, and a general formula for solving this problem is obtained.
Abstract: The classical secretary problem is generalized to admit stochastically successful procurement of previous interviewees, but each has a certain probability of refusing the offer. A general formula for solving this problem is obtained. Two special cases: constant probability of refusing and geometric probability of refusing are discussed in detail. The optimal stopping rules in these two cases turn out to be simple.
TL;DR: In this article, the limiting distributions of the actual waiting time and the virtual waiting time were determined for a single-server queue with Poisson input and general service times in the case where there are two types of services and no customer can stay in the system longer than an interval of length m.
Abstract: The limiting distributions of the actual waiting time and the virtual waiting time are determined for a single-server queue with Poisson input and general service times in the case where there are two types of services and no customer can stay in the system longer than an interval of length m.
TL;DR: In this paper, the quasi-stationary behavior of a Markov chain which is -irreducible when restricted to a subspace of a general state space is investigated. And it is shown that the existence of such limits as honest distributions is equivalent to the restricted chain being R-positive with the unique R-invariant measure satisfying a certain finiteness condition.
Abstract: The quasi-stationary behaviour of a Markov chain which is -irreducible when restricted to a subspace of a general state space is investigated. It is shown that previous work on the case where the subspaceis finite or countably infinite can be extended to general chains, and the existence of certain quasi-stationary limits as honest distributions is equivalent to the restricted chain being Rpositive with the unique R-invariant measure satisfying a certain finiteness condition.
TL;DR: In this article, the authors considered maximum likelihood estimators of the initial probabilities and the mean of a supercritical Galton-Watson process and deduced confidence intervals for these estimators, as well as for the Lotka-Nagaev estimator of the mean, the asymptotic distributions and deduce confidence intervals.
Abstract: In this article, we consider maximum likelihood estimators of the initial probabilities and the mean of a supercritical Galton-Watson process; we find for these estimators, as well as for the Lotka-Nagaev estimator of the mean, the asymptotic distributions and deduce confidence intervals. As these results hold even if the independence between individuals of the same generation is not satisfied, application to living populations may be considered.
TL;DR: In this paper, the authors made minor corrections to Heyde and Seneta (1972) and new convergence rate results given, as announced earlier, and new estimation by recurrence methods is discussed.
Abstract: Some minor corrections to Heyde and Seneta (1972) are made, and new convergence rate results given. Estimation by recurrence methods is discussed, as announced earlier. ESTIMATION THEORY; GROWTH AND IMMIGRATION RATES; MULTIPLICATIVE PROCESSES; STRONG CONSISTENCY; STRONG LAW; MARTINGALES; CENTRAL LIMIT THEOREM; FIRST ORDER AUTOREGRESSION; PARTICLE FLUCTUATION; LINDEBERG CONDITION; LAW OF ITERATED LOGARITHM; SMOLUCHOWSKI PROCESS; RECURRENCE ESTIMATION METHODS 1. Corrigenda to [3] (i) The constant y, introduced on p. 248, which is related to the constant B occurring in the statements of Theorem B and Theorem 3, by (1.1) B2 ya 2 + C4(1 n2 -1 should read as (1.2) y = (1-m3)-'[E(I-A)3 + jtE(Z1 )3 + 3mr2c2(L -m2)-1]. The definition of B on p. 237 should be changed accordingly. (Thanks are due to Miss T. Toe for bringing these typographical errors to our attention.) (ii) The dominating random variable X, such that (1.3) P(X > x) ? P(X > x), for all nt 0, x > 0, introduced on p. 241, is more clearly defined as follows. Let T(-) = min[n,n n 1: X, = 0], T(2) = min[n,n > 1: X,, = 01Xo = 0]. Define the random variables X(). X(2) respectively for the two cases by, X 0i ssn)) -0. j=1 Since s ~ nc2, and s2 as n -+ oo, this certainly holds if n-' n E(W2I(I w ) > Ssj)) o j=l as n -+ oo, and hence if (1.4) E(W,2I( WI > ss,)) -+ 0. Now w. = Y,E(Y, I1, ) = X,-mX,_, -: so that j W,I x) _ P(X, + ) > x) + P(Xn_ + A > x), i.e., (1.5) P(I WI > x) x) by (1.3) where EX2 ss,)) =j x2dP ( W, ss.)+ 2 xP(WI Wl > x)dx s~.) + L4 xP(X + 1 > x)dx -+0 as n -Xo, which is (1.4). This content downloaded from 207.46.13.133 on Sat, 17 Dec 2016 05:24:15 UTC All use subject to http://about.jstor.org/terms 574 C. C. HEYDE AND E. SENETA The justification for the second application is similar. (iv) In Section 5, while it was pointed out on p. 251 that the choice of time interval between observations is important for the branching process assumptions to be more closely satisfied for Smoluchowski's treatment of the particlefluctuation problem as a simple branching process with immigration, an erroneous impression is nevertheless given on p.252 that the actual underlying continuous-time process is the Markov branching process with immigratiom. This is not so in fact; the actual underlying continuous-time process (called the Smoluchowski process) is apparently not even Markovian, although the Markov branching process approximation is resorted to not infrequently. The reader is referred to various parts of the book of Bartlett [1], and also the book of Kac [4]. Ruben [7] nevertheless generalizes Smoluchowski's discrete-time treatment to what is, effectively, a very simple multi-type Galton-Watson process with immigration. One of the authors (E. S.) is most grateful to Professor Harold Ruben for a stimulating conversation on these lines. Several important contributions notwithstanding, the relation between discrete and continuous time discussed above remains somewhat shrouded in mystery; and an investigation by one of us with M. P. Quine is currently in progress.
TL;DR: In this paper, it was shown that a necessary and sufficient condition for the superposition of two ordinary renewal processes to again be a renewal process is that they be Poisson processes.
Abstract: Theorem: A necessary and sufficient condition for the superposition of two ordinary renewal processes to again be a renewal process is that they be Poisson processes. A complete proof of this theorem is given; also it is shown how the theorem follows from the corresponding one for the superposition of two stationary renewal processes.
TL;DR: In this paper, the demographic concept of average age at childbearing is given a rigorous interpretation and a unified treatment of the asymptotics is possible for a wide class, including for example the number of individuals having some random age dependent property or integrals of branching processes.
Abstract: With each individual in a branching population associate a random function of the age. Count the population by the values of these functions. Different choices yield different processes. In the supercritical case a unified treatment of the asymptotics is possible for a wide class, including for example the number of individuals having some random age dependent property or integrals of branching processes. As an application, the demographic concept of average age at childbearing is given a rigorous interpretation. ALMOST SURE CONVERGENCE; GENERAL BRANCHING PROCESSES
TL;DR: In this article, necessary and sufficient conditions for the existence of a proper non-degenerate limiting distribution for a Bellman-Harris age-dependent branching process with a compound renewal immigration component are found.
Abstract: Some necessary and sufficient conditions are found for the existence of a proper non-degenerate limiting distribution for a Bellman-Harris age-dependent branching process with a compound renewal immigration component. A number of these results are applicable to the batch arrival GI / G / ∞ queueing process. Some aspects of the situation when there is no such limiting distribution are considered. The situation when the immigration component is a nonhomogeneous compound Poisson process is briefly considered.
TL;DR: In this article, a stationary Poisson process of hyperplanes in Rn is characterized by the function 0 such that O(s) is the density of the Poisson point process induced on the straight lines with direction s. The set of these functions 0 is a convex cone o, a basis of which is a simplex e.
Abstract: A stationary Poisson process of hyperplanes in Rn is characterized (up to an equivalence) by the function 0 such that O(s) is the density of the Poisson point process induced on the straight lines with direction s. The set of these functions 0 is a convex cone o , a basis of which is a simplex e, and a given function 0 belongs to if and only if it is the supporting function of a symmetrical compact convex set which is a finite Minkowski sum of line segments or the limit of such finite sums. Another application is given concerning the tangential cone at h = 0 of a coveriance function. POISSON FLATS; SIMPLEX; NON ISOTROPIC COVARIANCE; COMPACT CONVEX SETS
TL;DR: The authors derived results for computing the first two moments of times in transient states and times to absorption in a transient semi-Markov process, and presented an illustrative example at the end.
Abstract: This paper derives results for computing the first two moments of times in transient states and times to absorption in a transient semi-Markov process. An illustrative example is presented at the end. SEMI-MARKOV PROCESS; FIRST TWO MOMENTS; TIMES IN TRANSIENT STATES
TL;DR: In this paper, an adaptive statistical procedure is proposed for which the expected payoff differs from the value by at most one-half plus a term which tends to zero (of order N −α, a N → ∞).
Abstract: Independent exponential capture times are assumed for each of N prey. The payoff is the number of prey caught less a linear time cost. The optimal stopping time and value are obtained. When N is known an adaptive statistical procedure is proposed for which the expected payoff differs from the value by at most one-half plus a term which tends to zero (of order N –α , a N → ∞.
TL;DR: In this paper, an optimal linear control for the symmetric team control problem specified by relations was deduced for the single-agent case, indicating that the recursions associated with effective state estimation are no longer of finite order.
Abstract: An optimal linear control (4) is deduced for the symmetric team control problem specified by relations (1)–(3) The generating function (6) for the optimal operator is in general irrational if there is more than one agent (controller), indicating that the recursions associated with effective state estimation are no longer of finite order, as in the single-agent case