Martingale Expansion in Mixed Normal Limit

The quasi-likelihood estimator and the Bayesian type estimator of the volatility parameter are in general asymptotically mixed normal. In case the limit is normal, the asymptotic expansion was derived in Yoshida (1997) as an application of the martingale expansion. The expansion for the asymptotically mixed normal distribution is then indispensable to develop the higher-order approximation and inference for the volatility. The classical approaches in limit theorems, where the limit is a process with independent increments or a simple mixture, do not work. We present asymptotic expansion of a martingale with asymptotically mixed normal distribution. The expansion formula is expressed by the adjoint of a random symbol with coefficients described by the Malliavin calculus, differently from the standard invariance principle. Applications to a quadratic form of a diffusion process ("realized volatility") are discussed.