Sankhya: The Indian Journal of Statistics

1995, Volume 57, Series A, Pt. 1, pp. 1--10

SUMS AND MAXIMA OF STATIONARY SEQUENCES WITH HEAVY TAILED DISTRIBUTIONS

By

C. W. ANDERSON, University of Sheffield

and

K. F. TURKMAN, University of Lisbon

SUMMARY.  We study the joint limiting distribution of sums and maxima of stationary sequences with distribution function which belongs to the domain of attraction (for sums) of a stable law of index $\alpha < 2$ . Under a week long-range dependence condition and a condition which rules out local cancellation of terms it is found that the joint limiting distribution is same as that for independent summands. An example shows that when local cancellation is possible the limiting dependence between sum and maximum may differ from that in the independence case. We give the main limiting result in two different forms : a random variable representation and a hybrid characteristic-distribution function representation.

AMS (1980) subject classification.  Primary 60F05; secondary 60G70.

Key words and phrases. Sums, maxima, hybrid characteristic-distribution function, stable law, extreme value distribution.

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