Asymptotic independence of three statistics of maximal segmental scores

Abstract

Let $ξ_1,ξ_2,\ldots$ be an iid sequence with negative mean. The $(m,n)$-segment is the subsequence $ξ_{m+1},\ldots,ξ_n$ and its \textit{score} is given by $\max\{\sum_{m+1}^nξ_i,0\}$. 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 $ξ_{1},\ldots,ξ_n$, 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 Cramér assumption on $ξ_1$, 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 $y=γ^{-1}\log n$ (where $γ$ is the Cramér coefficient), our results, together with the classical theorem of Iglehart \cite{Iglehart}, yield the existence and explicit form of the joint weak limit of $(R_n, R_n^* -y,O_{x+y})$.