Journal ArticleDOI

# The limiting distribution of the maximum term in a sequence of random variables defined on a markov chain.

01 Dec 1970-Journal of Applied Probability (Cambridge University Press)-Vol. 7, Iss: 3, pp 754-760
TL;DR: In this article, Gnedenko's classical work on the limit of the distribution of the maximum of a sequence of independent random variables is extended to the distribution that is defined on a finite Markov chain.
Abstract: : Gnedenko's classical work on the limit of the distribution of the maximum of a sequence of independent random variables is extended to the distribution of the maximum of a sequence of random variables defined on a finite Markov chain. (Author)
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TL;DR: In this article, the limit laws for the maxima of a sequence of random variables defined on a finite Markov chain which are conditionally independent given the chain are characterized for the case where the random variables are defined on the chain.
Abstract: : The limit laws are characterized for the maxima of a sequence of random variables defined on a finite Markov chain which are conditionally independent given the chain.

27 citations

### Cites background or methods from "The limiting distribution of the ma..."

• ...Equivalently Qn(x) = pn(x)M(x) + 0(1) where (2....

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• ...16) is applicable and lim Q?j(aijnX + bijn) = lim [pn(aijnx + bijn)Mij(aijnX + bijn)+ 0(1)], n--'oo n--'oo...

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• ...If p(x) is left continuous at Xo and Q(x) is irreducible for x > Xo - e for some e > 0, then Q(x) is left continuous at Xoo Proof· (1) If Q(x) is left continuous at Xo, select a sequence Xn t xo....

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• ...(1) We adopt the convention X o = - 00 instead of the more usual Xo=O because we deal with maxima rather than sums....

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• ...Therefore lim [pn(aijnX + bijn)Mij(aijnX + bijn) + 0(1)]....

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Journal ArticleDOI
TL;DR: In this article, it was shown that for independent identically distributed random variables where the distribution is a mixture, the convergence of their maxima is determined by one of the distributions in the mixture that has a dominant tail.
Abstract: A basic issue in extreme value theory is the characterization of the asymptotic distribution of the maximum of a number of random variables as the number tends to infinity. We address this issue in several settings. For independent identically distributed random variables where the distribution is a mixture, we show that the convergence of their maxima is determined by one of the distributions in the mixture that has a dominant tail. We use this result to characterize the asymptotic distribution of maxima associated with mixtures of convolutions of Erlang distributions and of normal distributions. Normalizing constants and bounds on the rates of convergence are also established. The next result is that the distribution of the maxima of independent random variables with phase type distributions converges to the Gumbel extreme-value distribution. These results are applied to describe completion times for jobs consisting of the parallel-processing of tasks represented by Markovian PERT networks or task-graphs. In these contexts, which arise in manufacturing and computer systems, the job completion time is the maximum of the task times and the number of tasks is fairly large. We also consider maxima of dependent random variables for which distributions are selected by an ergodic random environment process that may depend on the variables. We show under certain conditions that their distributions may converge to one of the three classical extreme-value distributions. This applies to parallel-processing where the subtasks are selected by a Markov chain.

18 citations

Journal ArticleDOI
TL;DR: This paper extended the results on extremal properties of chain-dependent sequences considered in Turkman and Walker (1983) by assuming conditions similar to those given by Leadbetter and Nandagopalan (1987) which permit clustering of high values.
Abstract: We extend the results on the extremal properties of chain-dependent sequences considered in Turkman and Walker (1983) by assuming conditions similar to those given by Leadbetter and Nandagopalan (1987) which permit clustering of high values.

9 citations

Journal ArticleDOI
TL;DR: In this paper, an inferential procedure for transformation models with conditional heteroscedasticity in the error terms is proposed, which is robust to covariate dependent censoring of arbitrary form.
Abstract: In this article we propose an inferential procedure for transformation models with conditional heteroscedasticity in the error terms. The proposed method is robust to covariate dependent censoring of arbitrary form. We provide sufficient conditions for point identification. We then propose an estimator and show that it is √n-consistent and asymptotically normal. We conduct a simulation study that reveals adequate finite sample performance. We also use the estimator in an empirical illustration of export duration, where we find advantages of the proposed method over existing ones.

7 citations

### Cites methods from "The limiting distribution of the ma..."

• ...The variance matrix can be consistently estimated by a numerical derivative form (see Sherman 1993 and Khan and Tamer 2007) or a kernel method....

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• ...Further assumptions imposed are analogous to those in Sherman (1993) to ensure the limiting objective function is sufficiently smooth (i.e., twice continuously differentiable) in β so we can work with its quadratic approximation....

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• ...Our limiting distribution theory for this estimator is based on the set of assumptions that are similar to those imposed in Sherman (1993) and Khan (2001)....

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Book ChapterDOI
01 Jan 1973
TL;DR: In this article, the authors present some of the basic results in the theory of order statistics, describe the trend of the work done in certain areas by referring to the landmark papers and indicate some recent results.
Abstract: : The objective of the present report is to present some of the basic results in the theory of order statistics, describe the trend of the work done in certain areas by referring to the landmark papers and indicate some of the recent results. After a brief description of the basic theory (Section 2) and the results concerning moments and inequalities (Section 3), some important asymptotic results of Gnedenko, Smirnov, Renyi, Berman and Kiefer have been discussed in Section 4. The next two sections deal with some applications of combinatorial methods in the theory of order statistics and fluctation theory. These results are mainly conerned with the applications of the ballot lemma and its generalizations and the use of the equivalence principle. Section 7 gives an outline of some problems in estimation and hypothesis testing. The last section discusses the role of order statistics in the subset selection problems and the algebraic structure involved in identification problems. (Author)

6 citations

##### References
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Book
01 Jan 1966
TL;DR: In this paper, the Basic Limit Theorem of Markov Chains and its applications are discussed and examples of continuous time Markov chains are presented. But they do not cover the application of continuous-time Markov chain in matrix analysis.
Abstract: Preface. Elements of Stochastic Processes. Markov Chains. The Basic Limit Theorem of Markov Chains and Applications. Classical Examples of Continuous Time Markov Chains. Renewal Processes. Martingales. Brownian Motion. Branching Processes. Stationary Processes. Review of Matrix Analysis. Index.

3,881 citations

Journal ArticleDOI

2,037 citations