Breaking multivariate records

For a sequence of i.i.d. d-dimensional random vectors with independent continuously distributed coordinates, say that the nth observation in sets record if it is not dominated every coordinate by an earlier observation; for j≤n, jth current at time n has been any first observations; and breaks k records there are observations n−1 but n. general dimension d, we identify, proof, asymptotic conditional distribution number broken given record. Fix let K(d) be variable this distribution. We show (right) tail satisfies P(K(d)≥k)≤exp−Ωk(d−1)∕(d2+d−3) as k→∞ P(K(d)≥k)≥exp−Ok1∕(d−1) k→∞. When d=2, description K(2) terms Poisson process agrees main result from Fill (2021) same G−1, where G∼Geometric(1∕2). Note lower bound on P(K(d)≥k) implies (shifted) Geometric d≥3. P(K(d)≥1)=exp[−Θ(d)] d→∞; particular, K(d)→0 probability d→∞.