Central limit theorems and parameter estimation associated with a weighted-fractional Brownian motion

•The paper studies Gaussian processes having not asymptotically stationary increments.•The results generalize partially some known results about fractional Brownian motion.•The paper constructs the estimator of the diffusion coefficient of an O-U model.•The proposed method is shown to be robust to a large class of Gaussian processes.

Let Ba,b be a weighted-fractional Brownian motion with indexes a and b satisfying |b|<1∧(1+a),a>−1 which is a central Gaussian process such that EBta,bBsa,b=1+b2∫0s∧tua((t−u)b+(s−u)b)du.In this paper, we consider the asymptotic normality associated with processes ∫0tBs+εa,b−Bsa,b2−taε1+bds,t∈[0,T],ε>0.As an application we study the asymptotic normality of the estimator of parameter σ>0 in stochastic process Xt=σBta,b−β∫0tXsds by using the generalized quadratic variation.