Estimating tail decay for stationary sequences via extreme values

Assaf Zeevi*, Peter W. Glynn

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


We study estimation of the tail-decay parameter of the marginal distribution corresponding to a discrete-time real-valued stationary stochastic process. Assuming that the underlying process is short-range dependent, we investigate properties of estimators of the tail-decay parameter which are based on the maximal extreme value of the process observed over a sampled time interval. These estimators only assume that the tail of the marginal distribution is roughly exponential, plus some modest 'mixing' conditions. Consistency properties of these estimators are established, as well as minimax convergence rates. We also provide some discussion on estimating the pre-exponent, when a more refined tail asymptotic is assumed. Properties of a certain moving-average variant of the extremal-based estimator are investigated as well. In passing, we also characterize the precise dependence (mixing) assumptions that support almost-sure limit theory for normalized extreme values and related first-passage times in stationary sequences.

Original languageEnglish
Pages (from-to)198-226
Number of pages29
JournalAdvances in Applied Probability
Issue number1
StatePublished - Mar 2004
Externally publishedYes


  • Almost-sure convergence
  • Estimation
  • Extreme values
  • Minimax
  • Strong mixing
  • Tail behavior
  • Uniform mixing


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