Breaking Multivariate Records

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 $d \to \infty$.