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

In: Mathematical Statistics and Probability Theory

Author

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  • George Haiman

    (Université Paris VI, I.N.R.E.T.S. and LSTA)

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.

Suggested Citation

  • George Haiman, 1987. "Almost Sure Asymptotic Behavior of the Record and Record Time Sequences of a Stationary Gaussian Process," Springer Books, in: M. L. Puri & P. Révész & W. Wertz (ed.), Mathematical Statistics and Probability Theory, pages 105-120, Springer.
    • Handle: RePEc:spr:sprchp:978-94-009-3963-9_10
    DOI: 10.1007/978-94-009-3963-9_10
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