Limits of bifractional Brownian noises

TL;DR

This paper investigates the asymptotic behavior of bifractional Brownian motion's increment process, showing convergence to a scaled fractional Brownian motion, and explores associated noise properties and limit theorems.

Contribution

It establishes the limit of the increment process of bifractional Brownian motion as the parameter h approaches infinity, revealing a connection to fractional Brownian motion.

Findings

Increment process converges to scaled fractional Brownian motion as h→∞

Behavior of bifractional Brownian noise characterized

Limit theorems for bifractional Brownian motion derived

Abstract

Let $B^{H,K}=(B^{H,K}_{t}, t\geq 0)$ be a bifractional Brownian motion with two parameters $H\in (0,1)$ and $K\in(0,1]$. The main result of this paper is that the increment process generated by the bifractional Brownian motion $(B^{H,K}_{h+t} -B^{H,K}_{h}, t\geq 0)$ converges when $h\to \infty$ to $(2^{(1-K)/{2}}B^{HK}_{t}, t\geq 0)$, where $(B^{HK}_{t}, t\geq 0)$ is the fractional Brownian motion with Hurst index $HK$. We also study the behavior of the noise associated to the bifractional Brownian motion and limit theorems to $B^{H,K}$.