Singular integrals of stable subordinator

It is well known that ∫01t−θdt<∞ for θ∈(0,1) and ∫01t−θdt=∞ for θ∈[1,∞). Since t can be taken as an α-stable subordinator with α=1, it is natural to ask whether ∫01t−θdSt has a similar property when St is an α-stable subordinator with α∈(0,1). We show that θ=1α is the border line such that ∫01t−θdSt is finite a.s. for θ∈(0,1α) and blows up a.s. for θ∈[1α,∞). When α=1, our result recovers that of ∫01t−θdt. Moreover, we give a pth moment estimate for the integral when θ∈(0,1α).