Intereting Posts

Conditionally combining vanilla and chocolate ice cream scoops
A complete orthonormal system contained in a dense sub-space.
Prove by induction $\sum_{i=0}^n i(i+1)(i+2) = (n(n+1)(n+2)(n+3))/4$
Global invertibility of a map $\mathbb{R}^n\to \mathbb{R}^n$ from everywhere local invertibility
Limit of quotients with square roots: $\lim_{x\to2} \frac{\sqrt{6-x}-2}{\sqrt{3-x}-1}$
Is Legendre’s solution of the general quadratic equation the only one?
Faster Convergence for the Smaller Values of the Riemann Zeta Function
Relation between independent increments and Markov property
how to solve the following mordell equation:$ y^2 = x^3 – 3$
Can $R \times R$ be isomorphic to $R$ as rings?
Generalization of “easy” 1-D proof of Brouwer fixed point theorem
Proving that an integral is differentiable
How to prove that the exponential function is the inverse of the natural logarithm by power series definition alone
what is the summation of such a finite sequence?
What are the generators for $\mathbb{Z}_p^*$ with p a safe prime?

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?

- Infinite divisibility of random variable vs. distribution
- From distribution to Measure
- $p$-variation of a continuous local martingale
- Expectation of maximum of arithmetic means of i.i.d. exponential random variables
- Cox derivation of the laws of probability
- A criterion for independence based on Characteristic function
- The finite-dimensional distributions of a centered Gaussian process are uniquely determined by the covariance function
- How common are probability distributions with a finite variance?
- Finitely Additive not Countably Additive on $\Bbb N$
- Time scaling of Brownian motion

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 $x=e^{-1}$.

$\hspace{4mm}$

**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)

$$

- Localising a polynomial ring and non-maximal prime ideal
- Why is UFD a Krull domain?
- Why are $3D$ transformation matrices $4 \times 4$ instead of $3 \times 3$?
- An Intuition to An Inclusion: “Union of Intersections” vs “Intersection of Unions”
- embdedding standard models of PA into nonstandard models
- If the empty set is a subset of every set, why isn't $\{\emptyset,\{a\}\}=\{\{a\}\}$?
- The Ext functor in the quiver representation
- Fake induction proofs
- $x^2, x^3$ being irreducible in $F$
- Binomial coefficients that are powers of 2
- What kind of compactness does “expanding $\mathbb{R}$ by constants” have?
- Comparing weak and weak operator topology
- Homeomorphism $\phi : T^2/A \to X/B$. What are $ T^2/A$ and $X/B$?
- Continuous function $f:\mathbb{R}\to\mathbb{R}$ such that $f(f(x)) = -x$?
- $f \in K$ of degree $n$ with Galois group $S_n$, why are there no non-trivial intermediate fields of $K \subset K(a)$ with $a$ a root of $f$?