Asymptotic independence of three statistics of maximal segmental scores

Let ξ1,ξ2,…ξ1,ξ2,… be an iid sequence with negative mean. The (m,n)(m,n)-segment is the subsequence ξm+1,…,ξnξm+1,…,ξn and its score   is given by max{∑m+1nξi,0}. Let RnRn be the largest score of any segment ending at time nn, Rn∗ the largest score of any segment in the sequence ξ1,…,ξnξ1,…,ξn, and OxOx the overshoot of the score over a level xx at the first epoch the score of such a size arises. We show that, under the Cramér assumption on ξ1ξ1, asymptotic independence of the statistics RnRn, Rn∗−y and Ox+yOx+y holds as min{n,y,x}→∞min{n,y,x}→∞. Furthermore, we establish a novel Spitzer-type identity characterising the limit law O∞O∞ in terms of the laws of (1,n)(1,n)-scores. As corollary we obtain: (1) a novel factorisation of the exponential distribution as a convolution of O∞O∞ and the stationary distribution of RR; (2) if y=γ−1logny=γ−1logn (where γγ 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).