I recently wrote a piece about the increasing polarization of the general public and why I believe it has a statistical basis in the sharp uptick in the number of opinions the average person is expected to hold. You can find that post here. I wanted to formalize the math behind the intuition but unfortunately Substack does not make it easy to write Latex so here we are.

There’s certain holes in my reasoning below that I’m still trying to make sense of. The ones I’m aware of I’ve elucidated, the others I hope will come in the form of feedback. Suggestions and corrections are always welcome.

We want to show that most randomly sampled Gaussian vectors in high dimensional spaces aren’t close to the center of the distribution but rather lie some distance away from the center. We also want to show what this distance is. I want to outline two possible approaches and show how they converge to (nearly) the same result:

1. Transformations between probability distributions combined with some neat tricks from convex optimization.
2. Brute force calculus.

Approach 1

Objective:

Show that a randomly sampled vector from an $n$ dimensional standard Gaussian will on average be $\sqrt{n}$ units of distance away from the origin.

Proof:

Consider a random vector $X$ sampled from a multivariate standard Gaussian.

$$ \mathbf{X} \sim \mathcal{N}(\boldsymbol{\mu}, \boldsymbol{\Sigma}),  \boldsymbol{\mu} = \mathbf{0} \quad \boldsymbol{\Sigma} = \mathbf{I} $$

We define $\Sigma$ as the identity covariance matrix in $R^{n}$ and $\mu$ as the zero vector. Each element of $X$ is drawn from the univariate standard normal distribution:

$$ x \sim \mathcal{N}(\boldsymbol{0}, \boldsymbol{1}) $$

Consider the squared length of this vector $X$ (length here refers to the L2 norm). We can write out this expression as: