Let k and m be two fixed positive integers with k[greater-or-equal, slanted]3 and 1<m<k. Let f(x1,x2,...,xk)=xk:m which is the mth smallest value of k real numbers x1,x2,...,xk. We call this f the mth order function of k real numbers x1,x2,...,xk. Let X0 be a nonconstant random variable with cumulative distribution F(x)=P(X0[less-than-or-equals, slant]x),x[set membership, variant](-[infinity],[infinity]). Let [xi] be the unique solution in (0, 1) to the equation B(x)=x where , which is the (cumulative) Beta distribution function with parameters m and k-m+1. Write [lambda]1=sup{x;F(x)<[xi]} and [lambda]2=inf{x;F(x)>[xi]}. Define the hierarchical sequence of random variables Xn,1,n[greater-or-equal, slanted]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[greater-or-equal, slanted]1} being independent random variables identically distributed as X0. In this note it is shown that and a.s. and a.s. It follows that a.s. if and only if [lambda]1=[lambda]2=[lambda]. 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].
