Sharpened localization of the trailing point of the Pareto record frontier

For $d\ge 2$ and iid $d$-dimensional observations $X^{(1)},X^{(2)},\dots$ 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 RS_n:=\{0\le x\in{\mathbb R}^d:x\not\prec X^{(i)}\ for all $1\le i\le n$\} at time $n$, where $0\le x$ means that $0\le x_j$ for $1\le j\le d$ and $x\prec y$ means that $x_j<y_j$ for $1\le j\le d$. With $x_{+}:={\sum }_{j=1}^d{x}_j$, let $$F_n^{-}:=\min \{x_{+}:x\in F_n\}\text{quadand}{F}_n^{+}:=\max \{x_{+}:x\in F_n\}\mathrm{.}$$ Almost surely, there are for each $n$ unique vectors ${\lambda }_n\in F_n$ and ${\tau }_n\in F_n$ such that $F_n^{+}=({\lambda }_n{)}_{+}$ and $F_n^{-}=({\tau }_n{)}_{+}$; we refer to ${\lambda }_n$ and ${\tau }_n$ 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 $\epsilon >0$ and $c_n\to \mathrm{\infty }$ we have $$\mathbb{P}(F_n^{-}-\ln n\in (-(2+\epsilon )\ln \ln \ln n,c_n))\to 1$$ (describing typical behavior) and almost surely $$\mathrm{lim\hspace{0.17em}sup}\frac{F_n^{-}-\ln n}{\ln \ln n}\le 0and\mathrm{lim\hspace{0.17em}inf}\frac{F_n^{-}-\ln n}{\ln \ln \ln n}\in [-2,-1]\mathrm{.}$$ 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 $$F_n^{-}-(\ln n-\ln \ln \ln n)-\ln (d-1)$$ (describing typical behavior) and, for $d\ge 3$, almost surely \begin{align*} &\limsup [F_n^- - (\ln n - \ln \ln \ln n)] \leq -\ln(d - 2) + \ln 2 \ \mbox{and }&\liminf [F_n^- - (\ln n - \ln \ln \ln n)] \ge - \ln d - \ln 2. \end{align*}