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?
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$\begingroup$ If $E[X_i]=\mu \not=0$, I would have thought your $T_1$ was likely to diverge as $n$ increases $\endgroup$– HenryCommented Jun 7, 2023 at 14:10
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$\begingroup$ Hi @Henry, thank you for your comment. I made a mistake in posting a question. I corrected it. Is your 1st comment still valid? If yes could you elaborate a little, how ? $\endgroup$– Jaimin ShahCommented Jun 7, 2023 at 19:28
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$\begingroup$ Yes - the first comment is still valid. Your $T_1$ will be about $\frac{\mu}{\sigma}\sqrt{n}$ which will not be bounded as $n$ increases $\endgroup$– HenryCommented Jun 7, 2023 at 19:44
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