Let g(x) = x/2 + 17/30 (mod 1), let 𝝃_i i = 1,2, … to be a sequence of independent, identically distributed random variables with uniform distribution on the interval [0,1/15], define g_i(x) = g(x) + 𝝃_i (mod 1) and for n = 1,2, …, define gⁿ (x) = g_n(g_n-1(…(g₁(x))…)). For x ϵ [0,1) let μ_n,x denote the distribution of gⁿ(x). The purpose of this note is to show that there exists a unique probability measure μ, such that, for all x ϵ [0,1); μ_n,x tends to μ as n → ∞. This contradicts a claim by Lasota and Mackey from 1987 stating that the process has an asymptotic three-periodicity.