On Multivariate Records from Random Vectors with Independent Components

TL;DR

This paper studies the properties of multivariate records from independent random vectors with independent components, revealing finite record counts in higher dimensions and analyzing the distribution and Markovian structure of complete records.

Contribution

It extends univariate record theory to multivariate vectors with independent components, showing finite record counts and characterizing the distribution of the terminal record.

Findings

Finite number of complete records for dimensions d ≥ 2

Distribution of the total number of complete records computed

Waiting times form a Markov chain with an absorbing state

Abstract

Let $\boldsymbol{X}_1,\boldsymbol{X}_2,\dots$ be independent copies of a random vector $\boldsymbol{X}$ with values in $\mathbb{R}^d$ and with a continuous distribution function. The random vector $\boldsymbol{X}_n$ is a complete record, if each of its components is a record. As we require $\boldsymbol{X}$ to have independent components, crucial results for univariate records clearly carry over. But there are substantial differences as well: While there are infinitely many records in case $d=1$, there occur only finitely many in the series if $d\geq 2$. Consequently, there is a terminal complete record with probability one. We compute the distribution of the random total number of complete records and investigate the distribution of the terminal record. For complete records, the sequence of waiting times forms a Markov chain, but differently from the univariate case, now the state infinity is an absorbing element of the state space.