The asymptotic distributions of sums of record values for distributions with lognormal-type tails

Abstract

LetX ₁,X ₂,/h. be a sequence of i.i.d. random variables and letX ⁽¹⁾ X ⁽²⁾,/h. be the associated record value sequence. We focus on the asymptotic distributions of sums of records,$T_n = \sum\nolimits_{k = 1}^n {X^{\left( k \right)} }$,forX ₁ ∈LN(γ). In this case, we find that 2 is a strange point for parameter γ. When γ > 2,T _n is asymptotically normal, while for 2 > γ > 1, we prove thatT _n cannot converge in distribution to any non-degenerate law through common centralizing and normalizing and logT _n is asymptotically normal.