Asymptotic independence of three statistics of maximal segmental scores

作者：Mijatovic Aleksandar; Pistorius Martijn^*

来源：Statistics & Probability Letters, 2015, 99: 185-191.

DOI：10.1016/j.spl.2015.01.015

• Maximal segmental score
• Asymptotic independence
• Asymptotic overshoot
• Random walk

Let xi(1), xi(2), ... be an lid sequence with negative mean. The (m, n)-segment is the subsequence xi(m+1), ..., xi(n) and its score is given by max{Sigma(n)(m+1) 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), ..., 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 Cramer assumption on xi(1), asymptotic independence of the statistics R-n, R-n*-y and Ox+y holds as min{n, y, x} -> infinity. Furthermore, we establish a novel Spitzer-type identity characterising the limit law O-infinity 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-infinity and the stationary distribution of R; (2) if y = gamma(-1) log n (where gamma is the Cramer coefficient), our results, together with the classical theorem of Iglehart (1972), yield the existence and explicit form of the joint weak limit of (R-n, R-n*-y, Ox+y).

• 出版日期2015-4