Weak convergence to multifractional Brownian motion of Riemann-Liouville type in Besov spaces

Abstract

We study the weak convergence of the family of processes {Vn(t)}n∈ℕ defined by $$V_n(t)=\int_{0}^t(t-u)^{H(t)-\frac{1}{2}}\theta_n(u)du,$$ where {θn(u)}n∈ℕ is a family of processes converging in law to a Brownian motion, as n→∞. We consider two cases of {θn}. First, we construct θn based on the well-known Donsker’s theorem and show that {Vn(t)}n∈ℕ converges in law to a multifractional Brownian motion of Riemann-Liouville type, as n→∞. Second, we construct θn based on a Poisson process, and then show that a multifractional Brownian motion of Riemann-Liouville type can be approximated in law by {Vn(t)}n∈ℕ.