A remark on stochastic monotonicity (Q1914311)

The following result provides (negative) information about the converse to a result of \textit{G. Samorodnitsky} and \textit{M. S. Taqqu} [Ann. Probab. 21, No. 1, 143-160 (1993; Zbl 0771.60017)] concerning the stochastic ordering \(X \geq_{\text{st}}Y\) where \(X\) and \(Y\) are infinitely divisible random vectors whose Lévy measures satisfy a regularity condition. The author proves the existence of random variables \(X\) and \(Y\) with compound Poisson distributions and Lévy measures \(\nu_X\) and \(\nu_Y\) concentrated on \([0, \infty)\) such that \(\forall x\) \(P(X \geq x) \geq P(Y \geq x)\) and for every \(a > 0\), \(\limsup_{x \to \infty} (\nu_Y ((x, \infty))/ \nu_X ((x/a, \infty))) = \infty\).