Hierarchical structures associated with order functions

Let k and m be two fixed positive integers with k⩾3 and 1ξ}. Define the hierarchical sequence of random variables Xn,1,n⩾0 by Xn+1,j=f(Xn,1+(j-1)k,Xn,2+(j-1)k,…,Xn,k+(j-1)k) with {X0,j,j⩾1} being independent random variables identically distributed as X0. In this note it is shown that Xn,1→dG(x)=ξI[λ1,∞)(x)+(1-ξ)I[λ2,∞)(x) and liminfn→∞Xn,1=λ1 a.s. and limsupn→∞Xn,1=λ2 a.s. It follows that limn→∞Xn,1=λ a.s. if and only if λ1=λ2=λ. This result generalizes and improves Propositions 4.3 and 4.4 of Li and Rogers [1999. Asymptotic behavior for iterated functions of random variables. Ann. Appl. Probab. 9, 1175-1201].