Singular integrals of stable subordinator

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