Small ball probabilities for a class of time-changed self-similar processes

TL;DR

This paper derives small ball probability estimates for a class of time-changed self-similar processes, revealing that decay rates depend on the self-similarity index rather than the outer process's small deviation order.

Contribution

It provides new small ball probability results for time-changed self-similar processes, including iterated fractional Brownian motion and inverse subordinators, highlighting the role of self-similarity.

Findings

Small ball probabilities decay with a power law.

Decay rate depends on the self-similarity index, not the outer process.

Results apply to processes like iterated fractional Brownian motion.

Abstract

This paper establishes small ball probabilities for a class of time-changed processes $X\circ E$, where $X$ is a self-similar process and $E$ is an independent continuous process, each with a certain small ball probability. In particular, examples of the outer process $X$ and the time change $E$ include an iterated fractional Brownian motion and the inverse of a general subordinator with infinite L\'evy measure, respectively. The small ball probabilities of such time-changed processes show power law decay, and the rate of decay does not depend on the small deviation order of the outer process $X$, but on the self-similarity index of $X$.