High level exceedances in stationary sequences with extremal index

Alpuim, M. Teresa

doi:10.1016/0304-4149(88)90073-7

Cited by 2 publications

(8 citation statements)

References 6 publications

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“…The atoms of this limiting process correspond to clusters of exceedances. Somewhat parallel results have also been obtained by Alpuim (1987).…”

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confidence: 74%

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A counterexample concerning the extremal index

Smith

1988

Adv. Appl. Probab.

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The concept of an extremal index, which is a measure of local dependence amongst the exceedances over a high threshold by a stationary sequence, has a natural interpretation as the reciprocal of mean cluster size.We exhibit a counterexample which shows that this interpretation is not necessarily correct. where 0~e~1. The parameter e was termed the extremal index by Leadbetter (1983), though the concept had occurred earlier in papers of Newell (1964), Loynes (1965), O'Brien (1974aO'Brien ( ), (1974b and Davis (1982). For a general overview of extremes in stationary sequences, see Leadbetter et ale (1983).It is possible to define an exceedance point process N; on (0, 1], such that Nn(s, t] is the number of exceedances of the level u; among {;r: ns < r~nt}. Convergence of {N n} as n~00 is studied by Hsing et al. (1988). One of their main results is that, if a limiting point process exists, then it must be compound Poisson. The atoms of this limiting process correspond to clusters of exceedances. Somewhat parallel results have also been obtained by Alpuim (1987).A natural interpretation of e is that 1/ e is the mean cluster size in the limiting point process. Hsing et al. were not, however, able to prove this without making additional assumptions. The following example shows that the result is false without such assumptions.The example is a regenerative sequence of the form (2)where (i) C j , j~1 are independent with a common distribution function F satisfying F(I) = 0, F(x) < 1 for all x < 00,(ii) For j> 1, given N1, ... '~-1' C1'···' C j with m~C j < m + 1, the probability of the event N, == i is qmi. Here {qmi' m~1, i~I} is a sequence of probabilities with qmi~0,In words, the process remains in state Ci for a random number of time epochs determined by the probability distribution qmi(i~1) with m = [C;], and then moves to a new state which is independently chosen from F.Let Pm = P{m~C j < m -I}, f.lm = I.iiqmi and suppose f.l = I.pmf.lm < 00. Then u is the mean recurrence time of the process. The process may be made stationary by a suitable choice of

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“…The atoms of this limiting process correspond to clusters of exceedances. Somewhat parallel results have also been obtained by Alpuim (1987).…”

supporting

confidence: 74%

“…The atoms of this limiting process correspond to clusters of exceedances. Somewhat parallel results have also been obtained by Alpuim (1987).A natural interpretation of e is that 1/ e is the mean cluster size in the limiting point process. Hsing et al were not, however, able to prove this without making additional assumptions.…”

supporting

confidence: 64%

A counterexample concerning the extremal index

Smith

1988

Adv. Appl. Probab.

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“…, X n if q n − j and put M (q) j,n := −∞ otherwise. For convenience, we write M (q) n for M (q) 0,n , M j,n for M (1) j,n and M n for M (1) 0,n . In the paper we investigate the asymptotic behaviour of M (q) n , as n → ∞, for {X n } with a regenerative structure.…”

Section: Introductionmentioning

confidence: 99%

“…. (we put S −1 := 0), representing regeneration times, such that for cycles C n := {X k : S n−1 k < S n } of the length Y n := S n − S n−1 and for ζ n denoting first r maxima in C n (we often write ζ n for ζ (1) n and put ζ (q) n := −∞ if q > Y n ), both of the following conditions hold:…”

Section: Introductionmentioning

confidence: 99%

“…Our goal is to describe the asymptotic behaviour of q-th maxima for regenerative processes. In order to do this, we combine the proof of Theorem 1 by Rootzén and the methods applied for stationary sequences by Hsing [5] (see also: Alpuim [1], Jakubowski [6]). In Section 2 we establish Theorem 2 and Corollary 3 describing the asymptotics of M (q) n .…”

Section: Introductionmentioning

confidence: 99%

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