Strong consistency of MLE for finite mixtures of location-scale distributions when the ratios of the scale parameters are exponentially small

In finite mixtures of location-scale distributions, if there is no constraint on the parameters then the maximum likelihood estimate does not exist. But when the ratios of the scale parameters are restricted appropriately, the maximum likelihood estimate exists. We prove that the maximum likelihood estimator (MLE) is strongly consistent, if the ratios of the scale parameters are restricted from below by $\exp(-n^{d}), 0 < d < 1 $, where $n$ is the sample size.