It is well known that f(0)(1) t(-theta)dt < infinity fore is an element of (0, 1) and f(0)(1) recit = infinity for theta is an element of [1, 00). Since t can be taken as an alpha-stable subordinator with alpha = 1, it is natural to ask whether f(0)(1); t(-theta)dS(t) has a similar property when S-t is an alpha-stable subordinator with a E (0, 1). We show that theta = 1/alpha is the border line such that integral(1)(0)t(-theta)dSt is finite a.s. for theta is an element of (0, 1/alpha) and blows up a.s. for theta is an element of [1/alpha, infinity). When alpha = 1, our result recovers that of f(0)(1)t(-theta) dt. Moreover, we give a pth moment estimate for the integral when theta is an element of (0, 1/alpha) (C) 2018 Elsevier B.V. All rights reserved.