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 partially removes a barrier in the applications of
the classical martingale limit theorem
to non-parametric estimates and inferences with non-stationarity,
and 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 improves
the existing works in literature.