Limit Theorems And Statistical Inference For Semimartingales

This dissertation focuses on the limit theorems and statistical inference for semimartingales. The main contents include the following aspects.Firstly, we obtain that locally square-integrable martingales and pure jump locally square-integrable martingales can be approximated by Guassian martin-gale in almost surely sense. The uniform approximate rates are obtained and the jumps are allowed in our limiting processes, moreover, the approximate rate can be determined by the jump parts of limiting processes.Secondly, the weak convergence of various general functionals of partial sums of causal process is obtained. We employ the martingale convergence method and martingale approximation for causal processes to get this result. We extend the results of Ibragimov and Phillips (2008) to more general case. The application of our result in the autoregression unit root theory is also given.Thirdly, the weak convergence of various general functionals of partial sums of i.i.d. heavy-tailed random variables is obtained.We employ the modified mar-tingale convergence method and stochastic calculus techniques to get this result. Our method is a new complement to the study of heavy tail phenomena.Fourthly, we consider the empirical likelihood inference for the jump-diffusion model. We construct the confidence intervals based on the empirical likelihood for the infinitesimal moments in the jump-diffusion models. They are better than the confidence intervals which are based on the asymptotic normality of point estimates.Finally, we present the local linear estimations for diffusion coefficient and drift coefficient in the second-order diffusion model. We show that under mild conditions, the estimators are weak consistent. We also use a Monte Carlo ex-periment to compare our estimators with the ones in Nicolau (2007).