Exceedance equivalence for increasing thresholds (${\mathrm{limsup}}_n\{{\max }_{k\le n}{X}_k>a_n\}={\mathrm{limsup}}_n\{X_n>a_n\}$)

Let $(a_n)$ be a nondecreasing real sequence tending to $+\infty$, and let $(x_n)$ be any real sequence. The running maximum exceeds $a_n$ infinitely often exactly when $x_n>a_n$ infinitely often. Applying this pathwise to random variables gives the displayed event equality, without any independence assumption.

One implication follows from $x_n\le {\max }_{k\le n}{x}_k$. For the other, if $x_n\le a_n$ from some $N$ onwards, every later value with $N\le k\le n$ satisfies $x_k\le a_k\le a_n$. The finite prefix also lies below $a_n$ for all sufficiently large $n$, because $a_n\to \infty$. Therefore the entire running maximum eventually stays below the threshold.

Divergence of the thresholds prevents one large early observation from creating infinitely many crossings. Monotonicity ensures that a later running-maximum crossing is controlled by the corresponding earlier individual threshold. These two conditions explain why the resulting event belongs to the tail sigma-algebra when the thresholds are deterministic.