On the convolution order of weak records

Two random variables are in convolution order, denoted X≤convYX≤convY, if there exists a random variable, called shift, T≥0T≥0, a.s., independent of X   such that Y=dX+T. Our main result states that for any shift T≥0T≥0 supported on the non-negative integers and any pair of integers 1≤n