Approximate distribution of product of N normal i.i.d.? Special case μ≈0

Given $N\geq30$ i.i.d. $X_n\approx\mathcal{N}(\mu_X,\sigma_X^2)$ , and $\mu_X \approx 0$ , looking for:

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

This is a special case $\mu_X \approx 0$ of a more general question.

— Andrei Pozolotin
fonte

1. Do you have any information about the

μ_{X}

$\mu_X$ and

σ_{X}

$\sigma_X$ ? (It would be nice if all

μ_{X} / σ_{X} ≫ 0

$\mu_X/\sigma_X \gg 0$ , for instance.) (2) An asymptotic normal approximation will be horrible, because asymptotically

Y

$Y$ will not look remotely normal.

— whuber

n

$n$

N (0, σ^{2})

$N(0, \sigma^2)$ . The non-zero

μ

$\mu$ case makes things much more complicated.

— wolfies

@whuber (1) after doing some monte carlo with some different

μ

$\mu$ and

σ

$\sigma$ , I found that distribution of

F

$F$ behaves rather well for

N > 30

$N>30$ and

| μ_{X} | \geq 10 σ_{X}

$|\mu_X|\geq10\sigma_X$ ; now I would like to find a nice expression for

μ_{F}

$\mu_F$ and

σ_{F}

$\sigma_F$ similar to how

χ^{2}

${\chi}^2$ has few nice approximations. I built few approximations via taylor expansion, but they misbehave badly. (2) well,

F

$F$ definitely "looks" like a sum of normal with chi squared, so

F

$F$ can be reduced to normal, if approximation "proves" that.

— Andrei Pozolotin

When

μ_{X} \geq 10 σ_{X}

$\mu_X \ge 10\sigma_X$ ,

Y

$Y$ will be nicely approximated by a lognormal distribution (as an application of the Barry-Esseen theorem to

\log (X)

$\log(X)$ shows).

— whuber

@whuber direct application of Barry-Esseen gives

F_{N} \approx 0 + \frac{1}{\sqrt{N}} Z

$F_N \approx 0 + \frac{1}{\sqrt{N}}Z$ , which is nice indeed, but it looses some structure:

μ_{F}

$\mu_F$ should be negative,

σ_{F}

$\sigma_F$ should depend on

α

$\alpha$ , etc. perhaps, there are better ways of applying it?

— Andrei Pozolotin

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 $f(x)$ :

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:

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

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

... 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 π)^{\frac{n}{2}} σ^{n}} MeijerG [{{}, {}}, {{0_{1}, \dots, 0_{n}}, {}}, \frac{x^{2}}{2^{n} σ^{2 n}}] for x \in R

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

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

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

— wolfies
fonte

Outstanding, thank you, Colin. Now I see why I must buy your book :-) Also makes me wonder if

l o g (. . . M e i j e r G (. . .))

$log(...MeijerG(...))$ looks any simpler. Time to dust off my Wolfram skills.

— Andrei Pozolotin