Domains of Geometric Partial Attraction of Max-Semistable Laws: Structure, Merge and Almost Sure Limit Theorems

Abstract

Max-semistable laws arise as non-degenerate weak limits of suitably centered and normed maxima of i.i.d. random variables along subsequences {k(n)}⊂ℕ such that k(n+1)/k(n)→c≥1, in which case the common distribution function F of the i.i.d. random variables is said to belong to the domain of geometric partial attraction of the max-semistable law. We give a necessary and sufficient condition for F to belong to the domain of geometric partial attraction of a max-semistable law and investigate the structure of these domains. We show that although weak convergence does not take place along {n}=ℕ, the distributions of the maxima “merge” together along the entire {n} with a suitably chosen family of limiting laws. The use of merge is demonstrated by almost sure limit theorems, which are also valid along the whole {n}.