Martingale limit theorems revisited and non-linear cointegrating regression

For a certain class of martingales, the convergence to mixture normal distribution is established under the convergence in distribution for the conditional variance. This is less restrictive in comparison with the classical martingale limit theorem where one generally requires the convergence in probability. The extension removes a main barrier in the applications of the classical martingale limit theorem to nonparametric estimates and inferences with non-stationarity, and essentially enhances the effectiveness of the classical martingale limit theorem as one of the main tools in the investigation of asymptotics in statistics, econometrics and other fields. The main result is applied to the investigations of asymptotics for the conventional kernel estimator in a nonlinear cointegrating regression, which essentially improves the existing works in literature.