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

We study the weak convergence of the family of processes { V n ( t )} n ∈ℕ defined by where { θ n ( u )} n ∈ℕ is a family of processes converging in law to a Brownian motion, as n →∞. We consider two cases of { θ n }. First, we construct θ n based on the well-known Donsker’s theorem and show that { V n ( t )} n ∈ℕ converges in law to a multifractional Brownian motion of Riemann-Liouville type, as n →∞. Second, we construct θ n based on a Poisson process, and then show that a multifractional Brownian motion of Riemann-Liouville type can be approximated in law by { V n ( t )} n ∈ℕ .