central limit theorem for a product

Given $-1\leq x_i\leq 1$ identically distributed random variables for $i=1,2,\dots n$. What is the distribution function of their product? Is there a central limit theorem for products if $n$ is large?

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The extension of the CLT to products would involve the $n^\text{th}$ root of $n$ variables. This raises problems when we consider random variables that might be negative. Therefore, let’s consider random variables $x_k\in[0,1]$ where $P(x_k\lt a)=F(a)$.

Let $u_k=\log(x_k)$, then $P(u_k\lt a)=F(e^a)$.

The mean of $u_k$ is
$$
\begin{align}
\mu
&=-\int_{-\infty}^0F(e^a)\,\mathrm{d}a\\
&=-\int_0^1F(t)\frac{\mathrm{d}t}{t}
\end{align}
$$
and the variance of $u_k$ is
$$
\begin{align}
\sigma^2
&=-2\int_{-\infty}^0aF(e^a)\,\mathrm{d}a-\left(\int_{-\infty}^0F(e^a)\,\mathrm{d}a\right)^2\\
&=-2\int_0^1F(t)\log(t)\frac{\mathrm{d}t}{t}-\left(\int_0^1F(t)\frac{\mathrm{d}t}{t}\right)^2
\end{align}
$$
So, if $-\int_0^1F(t)\log(t)\frac{\mathrm{d}t}{t}\lt\infty$, the standard CLT applies to $\log(x_k)$ and the $n^\text{th}$ root of the product of $n$ variables tends to
$$
e^{-\int_0^1F(t)\frac{\mathrm{d}t}{t}}
$$
Thus, the product of $x_k$ approximates a log-normal distribution where the log of the product has mean $n\mu$ and variance $n\sigma^2$. That is, the distribution of the $n^\text{th}$ root of the product of $n$ variables approximates
$$
\frac{\sqrt{n}}{x\sigma\sqrt{2\pi}}e^{-\frac{n}{2}\left(\frac{\log(x)-\mu}{\sigma}\right)^2}
$$
which tends to a Dirac delta at $x=e^\mu$.


The distribution of the logarithm of the product of $n$ of the $x_k$ will be the $n$-fold convolution of of the distribution of $\log(x_k)$, which is $e^aF'(e^a)$. The cumulative distribution of the logarithm of the product of $n$ of the $x_k$ is then
$$
F_n(e^a)=\overbrace{e^aF'(e^a)\ast e^aF'(e^a)\ast\dots\ast e^aF'(e^a)}^{n-1\text{ terms}}\ast F(e^a)
$$
The distribution of the product of $n$ of the $x_k$ is then $F_n’$


Example 1:
For a uniform distribution on $[0,1]$, we have $F(t)=t$ and the $n^\text{th}$ root of the product of $n$ variables tends to $e^{-1}$.

The distribution of the $n^\text{th}$ root of the product of $n$ uniform $[0,1]$ variables approximates
$$
\frac{\sqrt{n}}{x\sqrt{2\pi}}e^{-\frac{n}{2}(\log(x)+1)^2}
$$
which tends to a Diract Delta at $x=e^{-1}$.

$\hspace{4mm}$log-normal distributions

Example 2:
To compute the distribution of the product of two variables, we need to consider $F$ on a wider domain:
$$
F(t)=\left\{\begin{array}{l}
0&\text{if }t\lt0\\
t&\text{if }0\le t\le1\\
1&\text{if }t\gt1
\end{array}\right.
$$

we compute the convolution
$$
F_2(e^a)=e^aF'(e^a)\ast F(e^a)=(1-a)e^a
$$
Therefore, the cumulative distribution is
$$
F_2(t)=(1-\log(t))t
$$
and the distribution of the product of two uniformly distributed reals in $[0,1]$ is
$$
F_2′(t)=-\log(t)
$$