An introduction to a general records theory both for dependent and high dimensions

Abstract

The probabilistic investigation on record values and record times of a sequence of random variables defined on the same probability space has received much attention from 1952 to now. A great deal of such theory focused on iid or independent real-valued random variables. There exists a few results for real-valued dependent random variables. Some papers deal also with multivariate random variables. But a large theory regarding vectors and dependent data has yet to be done. In preparation of that, the probability laws of records are investigated here, without any assumption on the dependence structure. The results are extended sequences with values in partially ordered spaces whose order is compatible with measurability. The general characterizations are checked in known cases mostly for iid sequences. The frame is ready for undertaking a vast study of records theory in high dimensions and for types of dependence.

Keywords. partially ordered spaces, record values, record times, probability law, characterization of probability law.