The rates of convergence for generalized entropy of the normalized sums of IID random variables

We consider the generalized differential entropy of normalized sums of independent and identically distributed (IID) continuous random variables. We prove that the R\'{e}nyi entropy and Tsallis entropy of order $\alpha\ (\alpha>0)$ of the normalized sum of IID continuous random variables with bounded moments are convergent to the corresponding R\'{e}nyi entropy and Tsallis entropy of the Gaussian limit, and obtain sharp rates of convergence.