Breaking Multivariate Records

TL;DR

This paper analyzes the distribution of the number of records broken by observations in multivariate i.i.d. sequences, providing asymptotic bounds and characterizing the distribution for different dimensions.

Contribution

It identifies the asymptotic distribution of the number of records broken in multivariate sequences and shows that this distribution is not geometric for dimensions three and higher.

Findings

The tail bounds for the distribution of broken records are established.

For dimension 2, the distribution matches a shifted geometric distribution.

As dimension increases, the probability of breaking at least one record decreases exponentially.

Abstract

For a sequence of i.i.d. $d$-dimensional random vectors with independent continuously distributed coordinates, say that the $n$th observation in the sequence sets a record if it is not dominated in every coordinate by an earlier observation; for $j \leq n$, say that the $j$th observation is a current record at time $n$ if it has not been dominated in every coordinate by any of the first $n$ observations; and say that the $n$th observation breaks $k$ records if it sets a record and there are $k$ observations that are current records at time $n - 1$ but not at time $n$. For general dimension $d$, we identify, with proof, the asymptotic conditional distribution of the number of (Pareto) records broken by an observation given that the observation sets a record. Fix $d$, and let ${\mathcal K}(d)$ be a random variable with this distribution. We show that the (right) tail of ${\mathcal K}(d)$ satisfies \[ {\mathbb P}({\mathcal K}(d) \geq k) \leq \exp\left[ - \Omega\!\left( k^{(d - 1) / (d^2 + d - 3)} \right) \right]\ \ \mbox{as $k \to \infty$} \] and \[ {\mathbb P}({\mathcal K}(d) \geq k) \geq \exp\left[ - O\!\left( k^{1 / (d - 1)} \right) \right]\ \ \mbox{as $k \to \infty$}. \] When $d = 2$, the description of ${\mathcal K}(2)$ in terms of a Poisson process agrees with the main result from Fill [Comb. Probab. Comput. 30 (2021) 105--123] that ${\mathcal K}(2)$ has the same distribution as ${\mathcal G} - 1$, where ${\mathcal G} \sim \mbox{Geometric$(1/2)$}$. Note that the lower bound on ${\mathbb P}({\mathcal K}(d) \geq k)$ implies that the distribution of ${\mathcal K}(d)$ is NOT (shifted) Geometric for any $d \geq 3$. We show that ${\mathbb P}({\mathcal K}(d) \geq 1) = \exp[-\Theta(d)]$ as $d \to \infty$; in particular, ${\mathcal K}(d) \to 0$ in probability as $d \to \infty$.