# A Generalized Correlated Random Walk Converging to Fractional Brownian Motion

## TL;DR

This paper introduces a new algorithm for generating fractional Brownian motion with a specified Hurst parameter using correlated Bernoulli variables, establishing its convergence to the process through theoretical proof.

## Contribution

The authors develop a novel correlated Bernoulli-based algorithm for fractional Brownian motion generation and prove its convergence to the target Gaussian process.

## Key findings

- The proposed method accurately generates fractional Brownian motion with specified Hurst parameters.
- Theoretical proof confirms the convergence of the random walk to fractional Brownian motion.
- The algorithm leverages the correlation structure between Gaussian and Bernoulli variables.

## Abstract

We propose a new algorithm to generate a fractional Brownian motion, with a given Hurst parameter, 1/2<H<1 using the correlated Bernoulli random variables with parameter p; having a certain density. This density is constructed using the link between the correlation of multivariate Gaussian random variables and the correlation of their dichotomized binary variables and the relation between the correlation coefficient and the persistence parameter. We prove that the normalized sum of trajectories of this proposed random walk yields a Gaussian process whose scaling limit is the desired fractional Brownian motion with the given Hurst parameter, 1/2<H<1