A New Fine-Scale Berry-Esseen-Type Gumbel-Limit Theorem for Multivariate Maxima

Abstract

For d ≥ 2 and i.i.d. d-dimensional observations 𝐗^(1), 𝐗^(2), … with independent Exponential(1) coordinates, let φ_n denote the minimum 𝓁¹-norm among the maxima of {𝐗^(1), …, 𝐗^(n)}. (A maximum from this set is an observation 𝐗^(k) with 1 ≤ k ≤ n such that 𝐗^(k) ⊀ 𝐗^(i) for all 1 ≤ i ≤ n, where 𝐱 ≺ 𝐲 means that x_j < y_j for 1 ≤ j ≤ d.) Key roles in the study of multivariate Pareto records are played by φ_n and by the more easily handled maximum with the maximum 𝓁¹-norm. Fill et al. proved [Fill et al., 2026, Theorem 1.11(a)] that φ_n = ln n - ln ln ln n - ln(d - 1) + O_p(1/ln ln n), where Z_n = O_p(a_n) means that Z_n / a_n is bounded in probability, and conjectured [Fill et al., 2026, Remarks 1.13 and 3.3] that (ln ln n) (φ_n - [ln n - ln ln ln n - ln(d - 1)]) has a nondegenerate limiting distribution, suggesting that the limiting distribution might be that of - G, where G has a Gumbel distribution with location - ln[(d - 1)!]/(d - 1) and scale 1/(d - 1). In the present extended abstract we outline a proof of a Berry-Esseen-type theorem for this convergence in distribution, thereby establishing a very sharp result for φ_n.