Functional Transfer Theorems for Maxima of Stationary Processes

2000 Mathematics Subject Classification: 60G70, 60F12, 60G10.

In this paper we discuss the problem of finding the limit process of sequences of continuous time random processes, which are constructed as properly affine transformed maxima of random number identically distributed random variables. The max-increments of these processes are dependent. First we work under the well known conditions D (un) and D' (un) of Leadbetter, Lindgren and Rootzen, (1983). Further we investigate the case of moving average sequence. The distribution function of the noise components is assumed to have regularly varying tails or is subexponential and belongs to the max-domain of attraction of Gumbel distribution or belongs to the max-domain of attraction of Weibull distribution. We work with random time-components which are a.s. strictly increasing to infinity. In particular their counting process is a mixed Poisson process or a renewal process with regularly varying tails with parameter β ∈ (0, 1). Here is proved that such sequences of random processes converges weakly to a compound extremal process.

This research is partially supported by the NFSI, Grant VU-MI-105/2005 of the Ministry of Science and Education, Bulgaria.