It is well known that the distribution of the studentized mean, i.e., $T_0 = \frac{n^{1/2} (\bar{x}- \mu)}{\left(n^{-1} \sum \limits_{i=1}^N (x_i^2 - \bar{x})\right)^{-1 / 2}} $, can be approximated (in terms of the normal cdf/pdf) by Peter Hall theorem ("Edgeworth Expansion for Student’s t Statistic Under Minimal Moment Conditions"). Here is my question: Assume that we have very similar random variable, namely $T_1 = \frac{n^{1/2} (\bar{x})}{\left(n^{-1} \sum \limits_{i=1}^N (x_i^2 - \bar{x})\right)^{-1 / 2}} $ and the $x$'s are not zero mean; can we get a similar result, i.e., approximate the CDF of $T_1$ for any arbitrary i.i.d. distribution of the $x$'s?

It is well known that the distribution of the studentized mean, i.e., $T_0 = \frac{n^{1/2} (\bar{x}- \mu)}{\left(n^{-1} \sum \limits_{i=1}^n (x_i^2 - \bar{x}^2)\right)^{1 / 2}} $, can be approximated (in terms of the normal cdf/pdf) by Peter Hall theorem ("Edgeworth Expansion for Student’s t Statistic Under Minimal Moment Conditions"). Here is my question: Assume that we have very similar random variable, namely $T_1 = \frac{n^{1/2} (\bar{x})}{\left(n^{-1} \sum \limits_{i=1}^n (x_i^2 - \bar{x}^2)\right)^{1 / 2}} $ and the $x$'s are not zero mean; can we get a similar result, i.e., approximate the CDF of $T_1$ for any arbitrary i.i.d. distribution of the $x$'s?