Independence of the indicator functions of record values for Multivariate independent data

TL;DR

This paper rigorously proves that indicator functions of record values are independent for independent, not necessarily identically distributed, multivariate data in any dimension, extending previous proofs mainly focused on one dimension.

Contribution

It provides a detailed, rigorous proof of the independence of record indicator functions in multivariate independent data, generalizing beyond iid assumptions.

Findings

Indicator functions are independent for independent multivariate data.

Proof extends to non-iid data, broadening applicability.

Comparison with one-dimensional proofs highlights new generalization.

Abstract

We consider a sequence of random vectors on \(\mathbb{R}^d, \ d\geq 1\). We consider the record values based on the simultaneous strict inequality of the coordinates. The indicator record variable (irv) of the j-th observation is the function that assigns the value 1 (one) if that observation is a record value and the null value otherwise. Here, we give a detailed and a thorough proof that the indicator functions are independent, whenever the data are themselves independent, not necessarily <i>iid</i>, in \(\mathbb{R}^d, \ d\geq 1\). We compare that proof with available proofs in dimension one. Indeed, in seminal works on records, in particular in Ahsanullah(2024), Nevzorov(2001), Resnick (1987), Ahsanullah and Nevzorov (2015), etc., the independence of record indicator functions is usually validated based on logical reasoning, and so, is not rigorously proved. This allows us to undertake a detailed and thorough proof for independent data, not necessity <i>iid</i>, in \(\mathbb{R}^d, \ d\geq 1\). The proof includes leads for not even independent data.