On the convolution order of weak records

Abstract

Two random variables are in convolution order, denoted X≤convY, if there exists a random variable, called shift, T≥0, a.s., independent of X such that Y=dX+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 Wn≤convWm, with shift T, where Wk is the kth weak record from the parent X.