Strong consistency of MLE for finite mixtures of location-scale distributions when the scale parameters are exponentially small (full version)

In a finite mixture of location-scale distributions maximum likelihood estimator does not exist because of the unboundedness of the likelihood function when the scale parameter of some mixture component approaches zero. In order to study the strong consistency of maximum likelihood estimator, we consider the case that the scale parameters of the component distributions are restricted from below by $c_n$, where $c_n$ is a sequence of positive real numbers which tend to zero as the sample size $n$ increases. We prove that under mild regularity conditions maximum likelihood estimator is strongly consistent if the scale parameters are restricted from below by $c_{n} = \exp(-n^d)$, $0 < d < 1$.