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", Fₙ 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≤ xⱼ for 1≤ j≤ d and x y means that xⱼ<yⱼ for 1≤ j≤ d. With x₊:=∑ⱼ₌₁ᵈ xⱼ, let \[ F_n^-:=min\{x_+:x∈ F_n\} F_n^+:=max\{x_+:x∈ F_n\}. \] Almost surely, there are for each n unique vectors λₙ∈ Fₙ and τₙ∈ Fₙ such that Fₙ⁺=(λₙ)₊ and Fₙ⁻=(τₙ)₊; we refer to λₙ and τₙ 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 cₙ→∞ we have \[ { P}(F_n^- -ln n∈ (-(2+ε)lnlnln n,c_n))→ 1 \] (describing typical behavior) and almost surely \[ F_n^- - ln n/ln ln n ≤ 0 and F_n^- - ln n/ln ln ln n ∈ [-2, -1]. \] In this paper we use the theory of generators (minima of Fₙ) 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 {align*} & [F_n^- - (ln n - ln ln ln n)] ≤ -ln(d - 2) + ln 2 \\ and & [F_n^- - (ln n - ln ln ln n)] ≥ - ln d - ln 2. {align*}