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Given $N\geq30$ i.i.d. $X_n\approx\mathcal{N}(\mu_X,\sigma_X^2)$, and $\mu_X \approx 0$, 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 special case $\mu_X \approx 0$ of a more general question.

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  • $\begingroup$ 1. Do you have any information about the $\mu_X$ and $\sigma_X$? (It would be nice if all $\mu_X/\sigma_X \gg 0$, for instance.) (2) An asymptotic normal approximation will be horrible, because asymptotically $Y$ will not look remotely normal. $\endgroup$ – whuber Apr 3 '15 at 14:51
  • $\begingroup$ I just had a quick play with this. If you are interested, it is possible to obtain an exact closed form solution for the product of $n$ random variables that are iid $N(0, \sigma^2)$. The non-zero $\mu$ case makes things much more complicated. $\endgroup$ – wolfies Apr 3 '15 at 17:33
  • $\begingroup$ @whuber (1) after doing some monte carlo with some different $\mu$ and $\sigma$, I found that distribution of $F$ behaves rather well for $N>30$ and $|\mu_X|\geq10\sigma_X$; now I would like to find a nice expression for $\mu_F$ and $\sigma_F$ similar to how ${\chi}^2$ has few nice approximations. I built few approximations via taylor expansion, but they misbehave badly. (2) well, $F$ definitely "looks" like a sum of normal with chi squared, so $F$ can be reduced to normal, if approximation "proves" that. $\endgroup$ – Andrei Pozolotin Apr 3 '15 at 18:04
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    $\begingroup$ When $\mu_X \ge 10\sigma_X$, $Y$ will be nicely approximated by a lognormal distribution (as an application of the Barry-Esseen theorem to $\log(X)$ shows). $\endgroup$ – whuber Apr 3 '15 at 19:29
  • $\begingroup$ @whuber direct application of Barry-Esseen gives $F_N \approx 0 + \frac{1}{\sqrt{N}}Z$, which is nice indeed, but it looses some structure: $\mu_F$ should be negative, $\sigma_F$ should depend on $\alpha$, etc. perhaps, there are better ways of applying it? $\endgroup$ – Andrei Pozolotin Apr 4 '15 at 4:29
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It is possible to obtain an exact solution in the zero-mean case (part B).

The Problem

Let $(X_1, \dots, X_n)$ denote $n$ iid $N(0,\sigma^2)$ variables, each with common pdf $f(x)$:

enter image description here

We seek the pdf of $\prod_{i=1}^n X_i$, for $n = 2, 3, \dots$

Solution

The pdf of the product of two such Normals is simply:

enter image description here

... where I am using the TransformProduct function from the mathStatica package for Mathematica. The domain of support is:

enter image description here

The product of 3, 4, 5 and 6 Normals is obtained by iteratively applying the same function (here four times):

enter image description here

... where MeijerG denotes the Meijer G function

By induction, the pdf of the product of $n$ iid $N(0,\sigma^2)$ random variables is:


$$\frac{1}{(2 \pi )^{\frac{n}{2}} \sigma ^n} \text{MeijerG}[\{ \{ \}, \{ \} \}, \{ \{0_1, \dots, 0_n \}, \{ \} \}, \frac{x^2}{2^n \sigma ^{2 n}}] \quad \quad \text{ for } x \in \mathbb{R} $$


Quick Monte Carlo check

Here is a quick check comparing:

  • the theoretical pdf just obtained (when $n = 6$ and $\sigma=3$): RED DASHED curve
  • to the empirical Monte Carlo pdf: squiggly BLUE curve

enter image description here

Looks fine! [ the blue squiggly Monte curve is obscuring the exact red-dashed curve ]

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  • $\begingroup$ Outstanding, thank you, Colin. Now I see why I must buy your book :-) Also makes me wonder if $log(...MeijerG(...))$ looks any simpler. Time to dust off my Wolfram skills. $\endgroup$ – Andrei Pozolotin Apr 4 '15 at 17:59

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