On the Small Deviation Problem for Some Iterated Processes

TL;DR

This paper investigates the small deviation probabilities of iterated processes, specifically iterated Brownian motions and fractional Brownian motions, providing new results and a corrected proof of existing theorems.

Contribution

It offers general results on small deviations for iterated processes and calculates deviation rates for iterated Brownian motions and fractional Brownian motions, with a new proof of prior results.

Findings

Derived small deviation rates for iterated Brownian motions

Extended results to iterated fractional Brownian motions

Provided a new proof of existing laws of iterated logarithm

Abstract

We derive general results on the small deviation behavior for some classes of iterated processes. This allows us, in particular, to calculate the rate of the small deviations for $n$-iterated Brownian motions and, more generally, for the iteration of $n$ fractional Brownian motions. We also give a new and correct proof of some results in E. Nane, Laws of the iterated logarithm for $\alpha$-time Brownian motion, Electron. J. Probab. 11 (2006), no. 18, 434--459.