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Since $t$ can be taken as an $\\alpha$-stable subordinator with $\\alpha=1$, it is natural to ask whether $\\int_{0}^{1} t^{-\\theta} d S_{t}$ has a similar property when $S_{t}$ is an $\\alpha$-stable subordinator with $\\alpha \\in (0,1)$. We show that $\\theta=\\frac 1\\alpha$ is the border line such that $\\int_{0}^{1} t^{-\\theta} d S_{t}$ is finite a.s. for $\\theta \\in (0, \\frac 1\\alpha)$ and blows up a.s. for $\\theta \\in [\\frac1\\alpha,\\infty)$. 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