Sharpened Localization of the Trailing Point of the Pareto Record Frontier

Abstract

For d ≥ 2 and i.i.d. d-dimensional observations X^{(1)}, X^{(2)}, … with independent Exponential(1) coordinates, we revisit the study by Fill and Naiman (Electron. J. Probab., 25:Paper No. 92, 24 pp., 2020) of the boundary (relative to the closed positive orthant), or "frontier", F_n of the closed Pareto record-setting (RS) region RS_n := {0 ≤ x ∈ R^d: x ⊀ X^(i) for all 1 ≤ i ≤ n} at time n, where 0 ≤ x means that 0 ≤ x_j for 1 ≤ j ≤ d and x ≺ y means that x_j < y_j for 1 ≤ j ≤ d. With x_+ : = ∑_{j = 1}^d x_j = ‖x‖₁, let 
F_n^- := min{x_+: x ∈ F_n} and F_n^+ : = max{x_+: x ∈ F_n}. 
Almost surely, there are for each n unique vectors λ_n ∈ F_n and τ_n ∈ F_n such that F_n^+ = (λ_n)_+ and F_n^- = (τ_n)_+; we refer to λ_n and τ_n as the leading and trailing points, respectively, of the frontier. Fill and Naiman provided rather sharp information about the typical and almost sure behavior of F^+, but somewhat crude information about F^-, namely, that for any ε > 0 and c_n → ∞ we have 
P(F_n^- - ln n ∈ (- (2 + ε) ln ln ln n, c_n)) → 1 
(describing typical behavior) and almost surely 
limsup (F_n^- - ln n)/(ln ln n) ≤ 0 and liminf (F_n^- - ln n)/(ln ln ln n) ∈ [-2, -1]. 
In this extended abstract 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 - ln ln ln n) ⟶P -ln(d - 1) 
(describing typical behavior) and, for d ≥ 3, almost surely 
limsup [F_n^- -(ln n - ln ln ln n)] ≤ -ln(d - 2) + ln 2 
and liminf [F_n^- -(ln n - ln ln ln n)] ≥ -ln d - ln 2.