Limit distribution for point processes of high local maxima

Journal of Statistical Planning and Inference, 97(1) (2001), 227-233
http://dx.doi.org/10.1016/S0378-3758(00)00225-1

We define a local dependence condition which enables us to obtain a sufficient condition for the convergence in distribution of the sequence of point processes of high local maxima generated by a strictly stationary sequence of random variables. The limit point process is an homogeneous Poisson process. The result is applied to a stationary autoregressive sequence of maxima for which, after each upcrossing of a high level, we observe a downward tendency.