On the convolution order of weak records

Two random variables are in convolution order, denoted X ≤ conv Y , if there exists a random variable, called shift, T ≥ 0 , a.s., independent of X such that Y = d X + T . Our main result states that for any shift T ≥ 0 supported on the non-negative integers and any pair of integers 1 ≤ n < m there exists a discrete parent distribution X on the non-negative integers such that W n ≤ conv W m , with shift T, where Wk is the kth weak record from the parent X.