Let $\xi_1,\xi_2,\ldots$ be an iid sequence with negative mean. The
$(m,n)$-segment is the subsequence $\xi_{m+1},\ldots,\xi_n$ and its
\textit{score} is given by $\max\{\sum_{m+1}^n\xi_i,0\}$. Let $R_n$ be the
largest score of any segment ending at time $n$, $R^*_n$ the largest score of
any segment in the sequence $\xi_{1},\ldots,\xi_n$, and $O_x$ the overshoot of
the score over a level $x$ at the first epoch the score of such a size arises.
We show that, under the Cram\'er assumption on $\xi_1$, asymptotic independence
of the statistics $R_n$, $R_n^* -y$ and $O_{x+y}$ holds as
$\min\{n,y,x\}\to\infty$. Furthermore, we establish a novel Spitzer-type
identity characterising the limit law $O_\infty$ in terms of the laws of
$(1,n)$-scores. As corollary we obtain: (1) a novel factorization of the
exponential distribution as a convolution of $O_\infty$ and the stationary
distribution of $R$; (2) if $y=\gamma^{-1}\log n$ (where $\gamma$ is the
Cram\'er coefficient), our results, together with the classical theorem of
Iglehart \cite{Iglehart}, yield the existence and explicit form of the joint
weak limit of $(R_n, R_n^* -y,O_{x+y})$.