Summary
The class of non-degenerate joint limiting distributions for the maximum and minimum of stationary mixing sequences is determined. These limit distributions are of the form, H(x, ∞) −H(x, −y), where H(x,y) is a bivariate extreme value distribution. Unlike the classical result for i.i.d. sequences, the maximum and minimum of stationary mixing sequences may be asymptotically dependent. We also give a sufficient condition for the asymptotic independence of the maximum and minimum. Finally, an example of a stationary sequence satisfying the mixing condition D in Leadbetter but which is not weakly mixing is constructed.
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This research was supported in part by the National Science Foundation grant MCS 80-05483
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Davis, R.A. Limit laws for the maximum and minimum of stationary sequences. Z. Wahrscheinlichkeitstheorie verw Gebiete 61, 31–42 (1982). https://doi.org/10.1007/BF00537223
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DOI: https://doi.org/10.1007/BF00537223