Breaking multivariate records

Abstract

For a sequence of i.i.d. d-dimensional random vectors with independent continuously distributed coordinates, say that the nth observation in the sequence sets a record if it is not dominated in every coordinate by an earlier observation; for $j\le n$, say that the jth 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 nth 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 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

$$\mathbb{P}(\mathcal{K}(d)\ge k)\le exp\left[-\mathrm{\Omega }\left(k^{(d-1)∕(d^2+d-3)}\right)\right]\text{as}k\to \mathrm{\infty }$$

and

$$\mathbb{P}(\mathcal{K}(d)\ge k)\ge exp\left[-O\left(k^{1∕(d-1)}\right)\right]\text{as}k\to \mathrm{\infty }.$$

When $d=2$, the description of $\mathcal{K}(2)$ in terms of a Poisson process agrees with the main result from Fill (2021) that $\mathcal{K}(2)$ has the same distribution as $\mathcal{G}-1$, where $\mathcal{G}\sim \text{Geometric}(1∕2)$. Note that the lower bound on $\mathbb{P}(\mathcal{K}(d)\ge k)$ implies that the distribution of $\mathcal{K}(d)$ is not (shifted) Geometric for any $d\ge 3$.

We show that $\mathbb{P}(\mathcal{K}(d)\ge 1)=exp[-\mathrm{\Theta }(d)]$ as $d\to \mathrm{\infty }$; in particular, $\mathcal{K}(d)\to 0$ in probability as $d\to \mathrm{\infty }$.