Maxima and minima of complete and incomplete stationary sequences

Abstract

In the seminal contribution [R.A. Davis, Maxima and minima of stationary sequences, Ann. Probab. 7(3) (1979), pp. 453–460.] the joint weak convergence of maxima and minima of weakly dependent stationary sequences is derived under some mild asymptotic conditions. In this paper we address additionally the case of incomplete samples assuming that the average proportion of incompleteness converges in probability to some random variable . We show the joint weak convergence of the maxima and the minima of both complete and incomplete samples. It turns out that the maxima and the minima are asymptotically independent when is a deterministic constant.

Keywords::

• maxima
• minima
• incomplete sample
• joint limit distribution
• stationary sequences
• Berman condition
• Gaussian ARMA processes

2000 AMS Classification number::

• Primary 60G70
• Secondary 60G10

Acknowledgement

We would like to thank the referees for numerous comments and suggestions which significantly improved this contribution.

Additional information

Funding

E. Hashorva acknowledges support from the Swiss National Science Foundation [grant number 200021-140633/1]; Z. Weng has been supported by the Swiss National Science Foundation Project [grant number 200021-134785] and by the project RARE [grant number 318984] (a Marie Curie FP7 IRSES Fellowship).