Almost Sure Asymptotic Behavior of the Record and Record Time Sequences of a Stationary Gaussian Process

George Haiman
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George Haiman: Université Paris VI, I.N.R.E.T.S. and LSTA

A chapter in Mathematical Statistics and Probability Theory, 1987, pp 105-120 from Springer

Abstract: Abstract In (HAIMAN, 1985) we have shown, in the case of stationary Gaussian processes with a covariance function having rapidly decreasing tails, that the record and record times defined with respect to a threshold could be identified via a translation of the time index to the corresponding elements defined on an i.i.d. sequence. We show in this paper, by improving our method of proof, that the same result is true for the records and record times defined in the usual way. Namely, if {(Tn, θn), n ≥ 1} denotes the sequence of record times and records defined on the n given stationary sequence {X, n ≥ 1},we construct on the probability space on which are defined the Xn, possibly enlarged, a sequence {(Sn, Rn), n ≥ 1} such that: i) {(Sn,Rn), n ≥ 1} hãs the same probability law as {(Tn, θn), n ≥ 1} when the Xn are i.i.d., ii) thre exist a.s. n0 and q such tha? for any n ≥ n0 we have Sn =Tn-q and. Rn =θn-q.

Keywords: Asymptotic Behavior; Record Time; Stationary Sequence; Joint Density; Bernoulli Random Variable (search for similar items in EconPapers)
Date: 1987
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-94-009-3963-9_10

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DOI: 10.1007/978-94-009-3963-9_10

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