Singular integrals of stable subordinator

TL;DR

This paper investigates the finiteness of singular integrals involving stable subordinators, establishing a critical exponent for almost sure finiteness and providing moment estimates, extending classical results for standard integrals.

Contribution

It generalizes the classical integral finiteness criterion to stable subordinators, identifying the critical exponent and deriving moment estimates.

Findings

The integral is finite almost surely for  < heta < 1/\u03b1.

The integral diverges almost surely for   /.

Provides p-th moment estimates for the integral when  < heta < 1/.

Abstract

It is well known that $\int_{0}^{1} t^{-\theta} d t<\infty$ for $\theta \in (0,1)$ and $\int_{0}^{1} t^{-\theta} d t=\infty$ for $\theta \in [1,\infty)$. 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 $\theta \in [\frac1\alpha,\infty)$. When $\alpha=1$, our result recovers that of $\int_{0}^{1} t^{-\theta} d t$. Moreover, we give a $p$-th moment estimate for the integral when $\theta \in (0,\frac 1\alpha)$.