A note on the asymptotic distribution of the maximum likelihood estimator in a non-regular case

We consider the asymptotic distribution of the maximum likelihood estimator (MLE), when the log-likelihood ratio statistic weakly converges to the non-degenerated Gaussian process. We provide a simple expression for the density function of the asymptotic distribution by fundamental stochastic results. This note is helpful to investigate asymptotic properties of the MLE in a certain non-regular case.