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Given $N\geq30$ i.i.d. $X_n\approx\mathcal{N}(\mu_X,\sigma_X^2)$, and NO assumptions about $\mu_X$ and $\sigma_X$, looking for:

  1. accurate closed form distribution approximation of $Y_N=\prod\limits_{1}^{N}{X_n}$
  2. asymptotic (exponential?) approximation of same product

This is a general case, there are also some special cases:

A) $\mu_X \approx 0$

B) $\mu_X > 10 \sigma_X > 0$

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  • $\begingroup$ Please elaborate & clarify your post. Eg, is it a question? (There is no question mark.) What question are you asking, perchance? $\endgroup$ – gung - Reinstate Monica Apr 5 '15 at 15:49
  • $\begingroup$ this is relocation placeholder for first answer by wolfies, see original question and relocation request by whuber $\endgroup$ – Andrei Pozolotin Apr 5 '15 at 16:22
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    $\begingroup$ Please add whatever context is necessary to understand your question to this post. $\endgroup$ – gung - Reinstate Monica Apr 5 '15 at 16:39
  • $\begingroup$ The special cases are nearly meaningless, because the problem does not change when you change the units of measure of $X$. Thus condition (A) asserts nothing and condition $B$ merely asserts $\mu_X\ne 0$. $\endgroup$ – whuber Sep 11 '17 at 21:53