Limit theorem and LIL for some additive functionals associated to fBm and Riemann-Liouiville process

TL;DR

This paper establishes a strong approximation for the first order limit theorem of additive functionals related to fractional Brownian motion and Riemann-Liouville processes, and derives their law of iterated logarithm.

Contribution

It introduces a strong approximation version for the limit theorem of additive functionals of non-Markovian Gaussian processes, specifically fBm and Riemann-Liouville processes.

Findings

Established a strong approximation for the limit theorem.

Derived the law of iterated logarithm for the additive functionals.

Extended results to non-Markovian Gaussian processes.

Abstract

In this paper, we first establish a strong approximation version for the first order limit theorem of some additive functionals related to two non-Markovian Gaussian processes: the fractional Brownian motion (fBm) and the Riemann-Liouville process. As application, we give the law of iterated logarithm of the same additive functionals associated to this two processes