Showing papers on "Asymptotology published in 1979"
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TL;DR: In this paper, the authors analyze the recent development of the theory of the asymptotic distribution of extremes in the light of the questions (i) and (ii). Several dependence concepts will be introduced, each of which leads to a solution of (i).
Abstract: Abstract. Let X j denote the life length of the j th component of a machine. In reliability theory, one is interested in the life length Z n of the machine where n signifies its number of components. Evidently, Z n = min (X j : 1 ≤ j ≤ n). Another important problem, which is extensively discussed in the literature, is the service time W n of a machine with n components. If Y j is the time period required for servicing the j th component, then W n = max (Y j : 1 ≤ j ≤ n). In the early investigations, it was usually assumed that the X's or Y's are stochastically independent and identically distributed random variables. If n is large, then asymptotic theory is used for describing Z n or W n . Classical theory thus gives that the (asymptotic) distribution of these extremes (Z n or W n ) is of Weibull type. While the independence assumptions are practically never satisfied, data usually fits well the assumed Weibull distribution. This contradictory situation leads to the following mathematical problems: (i) What type of dependence property of the X's (or the Y's) will result in a Weibull distribution as the asymptotic law of Z n (or W n )? (ii) given the dependence structure of the X's (or Y's), what type of new asymptotic laws can be obtained for Z n (or W n )? The aim of the present paper is to analyze the recent development of the (mathematical) theory of the asymptotic distribution of extremes in the light of the questions (i) and (ii). Several dependence concepts will be introduced, each of which leads to a solution of (i). In regard to (ii), the following result holds: the class of limit laws of extremes for exchangeable variables is identical to the class of limit laws of extremes for arbitrary random variables. One can therefore limit attention to exchangeable variables. The basic references to this paper are the author's recent papers in Duke Math. J. 40 (1973), 581–586, J. Appl. Probability 10 (1973, 122–129 and 11 (1974), 219–222 and Zeitschrift fur Wahrscheinlichkeitstheorie 32 (1975), 197–207. For multivariate extensions see H. A. David and the author, J. Appl. Probability 11 (1974), 762–770 and the author's paper in J. Amer. Statist. Assoc. 70 (1975), 674–680. Finally, we shall point out the difficulty of distinguishing between several distributions based on data. Hence, only a combination of theoretical results and experimentations can be used as conclusive evidence on the laws governing the behavior of extremes.
1,964 citations
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TL;DR: In this article, the authors extended Aronson's and Weinberger's concept of asymptotic speed to the nonlinear integral equation t J g(u(t-s,x+y))k(s9\\y\\)dsdy, which is a spatial version of Kermack's and McKendrick's epidemic model.
Abstract: Recently Aronson [1] extended the concept of asymptotic speed which he and Weinberger [3], [4] had developed for nonlinear diffusion problems in population genetics, combustion and nerve propagation, to an epidemic model proposed by Kendall [11], [12] in 1957 (1965). In this model (which is a spatial version of Kermack's and McKendrick's epidemic model [13]) the aflfected individuals become immediately infectious and are removed at a constant rate. The model does not take into account that with most infectious diseases the affected individuals underlie an incubation period, before they become infective, and that they remain infective for a fixed period only. These features cannot be described by the equation considered by Aronson [1] which contains a derivative with respect to time and an integral with respect to space. It is therefore desirable to extend Aronson's and Weinberger's concept of asymptotic speed to the nonlinear integral equation t J g(u(t-s,x+y))k(s9\\y\\)dsdy
159 citations
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20 citations
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17 citations
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12 citations
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01 Jan 1979TL;DR: In this article, the asymptotic behavior in K of the Kth order of the perturbative expansion has been investigated with remarkable success for both interacting bosons and fermions.
Abstract: During last year, the asymptotic behaviour in K of the Kth order of the perturbative expansion has been widely investigated with remarkable success1–3. If the interaction is superrenormalizable, the problem has been solved for both interacting bosons4, 5 and fermions6–8: of course technical details may change from theory to theory. When the interaction is renormalizable new difficulties arise9, 10; they are not completely understood at the present moment.
9 citations
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6 citations
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5 citations
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01 Sep 1979
TL;DR: In this article, the asymptotic distribution of a U-statistic is found in the case when the corresponding Von Mises functional is stationary of order 1, and practical methods for tabulation of the limit distributions are discussed.
Abstract: : The asymptotic distribution of a U-statistic is found in the case when the corresponding Von Mises functional is stationary of order 1. Practical methods for the tabulation of the limit distributions are discussed, and the results extended to certain incomplete U-statistics. (Author)
5 citations
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TL;DR: In this article, the authors deal with the problem of constructing and proving asymptotic expansions for nonlinear, singularly perturbed difference equations, based on the concepts of e-stability and matching.
Abstract: The present paper deals with the problem of constructing and proving asymptotic expansions for nonlinear, singularly perturbed difference equations. New methods for the construction of asymptotic expansions are presented and compared with well-known ones. For the proof of their validity, fundamental principles for the treatment of nonlinear singular perturbation problems are applied, based on the concepts of e-stability, formal asymptotic expansions, matching and asymptotic expansions. The results are derived from a general theory of asymptotic expansions of nonlinear operator equations that has been developed recently by the author.
5 citations
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01 Jan 1979••
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TL;DR: In this article, the second paper of the series concerning the renormalized G -convolution product H G ren in the Euclidean space associated with a general graph G with n external lines is presented.
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TL;DR: In this article, the authors derived mathematical conditions which characterize the asymptotic behavior of a class of finite difference equations and applied them to the theory of optimal control, and showed that these conditions find application in optimal control.
Abstract: Mathematical conditions are derived which characterize the asymptotic behaviour of a. class of finite difference equations. The results find application in the theory of optimal control.
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TL;DR: In this article, a continuation method is applied to a singular perturbation parameter to obtain a new numerical method for computing optimal trajectories, which allows one to use simply calculated reduced order approximations as starting solutions and continue the perturbations until the optimal full-order solution is obtained.
Abstract: A continuation method is applied to a singular perturbation parameter to obtain a new numerical method for computing optimal trajectories. This method allows one to use simply calculated reduced order approximations as starting solutions and continue the perturbation parameter until the optimal full-order solution is obtained. The method does not require the calculation of higher order correction terms nor does it require the perturbation parameter to be small - thus, it has potentially superior convergence properties compared to conventional asymptotic expansions when the perturbation parameter is large. A simple trajectory optimization problem is considered to illustrate the method.
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