  Central limit theorems play an important role in the study of statistical
inference for stochastic processes. However, when the nonparametric local
polynomial threshold estimator, especially local linear case, is employed to
estimate the diffusion coefficients of diffusion processes, the adaptive and
predictable structure of the estimator conditionally on the $\sigma-$field
generated by diffusion processes is destroyed, the classical central limit
theorem for martingale difference sequences can not work. In this paper, we
proved the central limit theorems of local polynomial threshold estimators for
the volatility function in diffusion processes with jumps. We believe that our
proof for local polynomial threshold estimators provides a new method in this
fields, especially local linear case.
