An introduction to a general records theory both for dependent data and high dimensions: revisited 2

TL;DR

This paper revisits the probabilistic theory of record values and times, extending it to dependent, high-dimensional data without assuming independence, and lays groundwork for future research in complex data structures.

Contribution

It develops a general framework for records in dependent and high-dimensional data, filling gaps in existing theory that mainly focuses on independent real-valued variables.

Findings

Probability laws of records are characterized without dependence assumptions.

Results are extended to sequences in partially ordered spaces.

Framework prepares for future high-dimensional and dependent data record analysis.

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 \textit{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 \textit{iid} sequences. The frame is ready for undertaking a vast study of records theory in high dimensions and for types of dependence.