Sharpened localization of the trailing point of the Pareto record frontier

For d≥2 and iid d-dimensional observations X^(1),X^(2),… with independent Exponential(1) coordinates, we revisit the study by Fill and Naiman (Electron. J. Probab., 2020) of the boundary (relative to the closed positive orthant), or "frontier", F_n of the closed Pareto record-setting (RS) region _n:={0≤ x∈ℝ^d:x⊀X^(i)} at time n, where 0≤ x means that 0≤ x_j for 1≤ j≤ d and x≺ y means that x_j0 and c_n→∞ we have ℙ(F_n^- -ln n∈ (-(2+ε)lnlnln n,c_n))→ 1 (describing typical behavior) and almost surely lim supF_n^- - ln n/lnln n≤ 0 lim infF_n^- - ln n/lnlnln n∈ [-2, -1]. In this paper we use the theory of generators (minima of F_n) together with the first- and second-moment methods to improve considerably the trailing-point location results to F_n^- - (ln n - lnlnln n) P⟶ - ln(d - 1) (describing typical behavior) and, for d ≥ 3, almost surely lim sup [F_n^- - (ln n - lnlnln n)] ≤ -ln(d - 2) + ln 2 lim inf [F_n^- - (ln n - lnlnln n)] ≥ - ln d - ln 2.