Various asymptotically correct bounds on the uniform metric for distance between distribution functions in the central limit theorem for sums of independent and identically distributed random variables have previously been given. It is shown in the present paper that corresponding nonuniform bounds can be given for the difference between distribution functions. These results have much wider applicability, such as for obtaining probabilities of moderate deviation or for dealing with $L_p$ metrics, $1 \leqq p \leqq \infty$.
"A Nonuniform Bound on Convergence to Normality." Ann. Probab. 3 (5) 903 - 907, October, 1975. https://doi.org/10.1214/aop/1176996280