Convergence of tail sum for records

Abstract

Suppose {R_n^(L) (F): n≥1} is the sequence of lower records from a distribution F, where F is continuous with inf {x ∈ supp(F)}=0. We derive conditions under which logarithm of the tail sum of records, ∑ _j=n ^∞=R_n ^(L) (F), properly centered and scaled, converge weakly. We also prove two results on ?-varying and regularly varying functions, which are of independent interest.