2014/02/24 by Aleksandar Mijatović, Mijatović, Aleksandar, Martijn Pistorius +1
Computer Science · Decision Sciences · Economics, Econometrics and Finance · Mathematics · #60G51 #Bayesian Methods and Mixture Models #FOS: Mathematics #Financial Risk and Volatility Modeling #Probability (math.PR) #Probability and Risk Models #math.PR #msc:60G51
paper · pdf · doi:10.48550/arxiv.1402.5858
13 pages, no figures
arxiv created 2014/02/24 · openalex publication_date 2014/02/24 · arxiv updated 2014/02/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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, R^*n 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 factorization of the exponential distribution as a convolution of O_∞ and the stationary distribution of R; (2) if y=γ-1log n (where γ is the Cramér coefficient), our results, together with the classical theorem of Iglehart \citeIglehart, yield the existence and explicit form of the joint weak limit of (Rn, Rn^* -y,Ox+y).