Before revision:  Besides the widely-used Kullback-Leibler (KL) divergence which requires the distribution of the objective function to be exponentially bounded, our results apply to divergence measures that accommodate well heavy tail distribution such as the student t-distribution and the lognormal distribution . Comprehensive testings on synthetic data and real data are provided .
After revision:  Besides the widely-used Kullback-Leibler (KL) divergence which requires the distribution of the objective function to be exponentially bounded, our high-order expansion result, a notable feature of our analysis is that it applies to divergence measures that accommodate well heavy tail distribution such as the student t-distribution and the lognormal distribution . Comprehensive testings on synthetic data and real data are provided .