On the renewal measure for Gaussian sequences

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A form for U(t), the expected number of times a Gaussian sequence falls below a level of t, is given in terms of the mean M(x) and the variance V2(x) functions. It is shown that under general conditions U(t) ∼ M(-1)(t), t → ∞. Moreover, if M and V are regularly varying at infinity functions, then U(t) - M(-1)(t) is also regularly varying at infinity. A renewal theorem for stationary Gaussian sequences is given, where it is shown that the asymptotic behavior of U(t) - t/μ is determined by the asymptotic behavior of V2(t)/t.

Original languageEnglish
Pages (from-to)167-171
Number of pages5
JournalStatistics and Probability Letters
Issue number4
Publication statusPublished - 1 Jan 1986
Externally publishedYes


  • Gaussian sequence
  • renewal theorem
  • stationary sequence

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