Extremes of independent exponential variables
= Extremes of independent exponential variables
For independent rate-one exponential variables $X_n$ and $M_n=\max_{m\leq n}X_m$,
$$
\limsup_{n\to\infty}\frac{X_n}{\log n}=1,
\qquad
\liminf_{n\to\infty}\frac{M_n}{\log n}\geq1
$$
almost surely. The first follows by applying both Borel--Cantelli lemmas to thresholds $(1\pm\varepsilon)\log n$; the second uses
$\mathbb P(M_n\leq(1-\varepsilon)\log n)\leq e^{-n^\varepsilon}$.