Asymptotic independence of three statistics of maximal segmental scores

Let ξ1,ξ2,… be an iid sequence with negative mean. The (m,n)-segment is the subsequence ξm+1,…,ξn and its score is given by max{∑m+1nξi,0}. Let Rn be the largest score of any segment ending at time n, Rn∗ the largest score of any segment in the sequence ξ1,…,ξn, and Ox 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 ξ1, asymptotic independence of the statistics Rn, Rn∗−y and Ox+y holds as min{n,y,x}→∞. Furthermore, we establish a novel Spitzer-type identity characterising the limit law O∞ in terms of the laws of (1,n)-scores. As corollary we obtain: (1) a novel factorisation of the exponential distribution as a convolution of O∞ and the stationary distribution of R; (2) if y=γ−1logn (where γ is the Cramér coefficient), our results, together with the classical theorem of Iglehart (1972), yield the existence and explicit form of the joint weak limit of (Rn,Rn∗−y,Ox+y).